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default +\quotes_language english +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +More Reductions +\end_layout + +\begin_layout Author +Nike Dattani +\end_layout + +\begin_layout Author Email + +n.dattani@cfa.harvard.edu +\end_layout + +\begin_layout Affiliation +Harvard-Smithsonian Center for Astrophysics +\end_layout + +\begin_layout Standard +\begin_inset VSpace defskip +\end_inset + + +\end_layout + +\begin_layout Section +Theorem 4.1 of Anthony-Boros-Crama-Gruber (ABCG) +\end_layout + +\begin_layout Standard +Any symmetric fuction can be quadratized with +\begin_inset Formula $n-2$ +\end_inset + + auxiliaries, where +\begin_inset Formula $\alpha_{i}$ +\end_inset + + comes from Corollary 2.3: +\end_layout + +\begin_layout Standard + +\size footnotesize +\begin_inset Formula +\begin{align} +f\left(b_{1},b_{2},\ldots,b_{n}\right) & \rightarrow-\alpha_{0}-\alpha_{0}\sum_{i}b_{i}+2a_{2}\sum_{ij}b_{i}b_{j}+\\ + & 2\sum_{2i-1}\left(\alpha_{2i-1}-{\rm min}\left(\alpha_{2j-1}\right)\right)b_{a_{2i-1}}\left(2i-\frac{3}{2}-\sum_{j}b_{j}\right)+\\ + & 2\sum_{2i}\left(\alpha_{2i}-{\rm min}\left(\alpha_{2j}\right)\right)b_{a_{2i}}\left(2i-\frac{1}{2}-\sum_{j}b_{j}\right)\\ +\alpha_{i} & =-4\sum_{j=0}^{i}\left(-1\right)^{i-j}f(j)-f(i-1)+3f(i) +\end{align} + +\end_inset + + +\size default +Alternatively: +\end_layout + +\begin_layout Standard + +\size footnotesize +\begin_inset Formula +\begin{align} +f\left(b_{1},b_{2},\ldots,b_{n}\right) & \rightarrow-\alpha_{0}-\alpha_{0}\sum_{i}b_{i}+a_{2}\sum_{ij}b_{i}b_{j}+2\sum_{i}\left(\alpha_{i}-c\right)b_{a_{i}}\left(2i-\frac{1}{2}-\sum_{j}b_{j}\right)\\ +c & =\begin{cases} +{\rm min}\left(\alpha_{2j}\right) & ,i\in\text{even}\\ +{\rm min}\left(\alpha_{2j-1}\right) & ,i\in\text{odd} +\end{cases}\\ +a_{2} & =\text{has to be obtained from page 12 of the paper.} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard + +\color red +They say this is linear in the auxiliary variables, but it doesn't seem + to be, because we have +\begin_inset Formula $b_{a_{i}}b_{j}$ +\end_inset + + terms where +\begin_inset Formula $b_{a_{i}}$ +\end_inset + + are auxiliaries. +\end_layout + +\begin_layout Standard +Pro: quadratization symmetric in all non-auxiliary variables, which isn't + true for all quadratizations of symmetric functions. + Reproduces the full spectrum. +\end_layout + +\begin_layout Standard +Con: all quadratic terms of the non-auxiliary variables, are non-submodular. + Also very complicated and uses more auxiliaris than simlper methods. +\end_layout + +\begin_layout Section +Unpublished work of Alexander Fix +\end_layout + +\begin_layout Standard +Any symmetric fuction can be quadratized with +\begin_inset Formula $n-1$ +\end_inset + + auxiliaries. + Add a multiple of +\begin_inset Formula $E(\sum b_{r})$ +\end_inset + + to each term of Corollary 2.4 of the above paper. + +\end_layout + +\begin_layout Section +Asymmetric reduction for negative monomials of arbitrary +\begin_inset Formula $k$ +\end_inset + + +\end_layout + +\begin_layout Standard + +\size footnotesize +\begin_inset Formula +\begin{align} +-b_{1}b_{2}\ldots b_{k} & \rightarrow(k-1)b_{k}b_{a}-\sum_{i}b_{i}(b_{a}+b_{k}-1)\\ +-b_{1}b_{2}\ldots b_{k} & \rightarrow-\sum_{i}b_{i}-\sum_{i}b_{i}b_{k}-\sum_{i}b_{i}b_{a}+(k-1)b_{k}b_{a} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +Pro: only 1 auxiliary to quadratize k degree term. + Only one non-submodular term (and it's quadratic). + Reproduces the full spectrum. +\end_layout + +\begin_layout Standard +Con: Turns symmetric into non-symmetric (but only the +\begin_inset Formula $b_{k}$ +\end_inset + + is asymmetric). +\end_layout + +\begin_layout Section +\begin_inset Quotes eld +\end_inset + +A related paper by the present authors [1] gives a complete characterization + of all the quadratizations of negative monomials involving one auxiliary + variable +\begin_inset Quotes erd +\end_inset + + +\end_layout + +\begin_layout Section +Another reduction for negative monomials of arbitrary +\begin_inset Formula $k$ +\end_inset + + +\end_layout + +\begin_layout Standard + +\size footnotesize +\begin_inset Formula +\begin{align} +-b_{1}b_{2}\ldots b_{k} & \rightarrow2b_{a}\left(k-\frac{1}{2}-\sum_{i}b_{i}\right)\\ +-b_{1}b_{2}\ldots b_{k} & \rightarrow\left(2k-1\right)b_{a}-2\sum_{i}b_{i}b_{a} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +Pro: only 1 auxiliary to quadratize k degree term. + Only one non-submodular term (and it's linear). + Symmetric with respect to all non-auxiliary variables. + Reproduces the full spectrum. +\end_layout + +\begin_layout Standard +Con: Coefficients of quadratic terms are twice the size of in the +\begin_inset Quotes eld +\end_inset + +standard +\begin_inset Quotes erd +\end_inset + + quadratization for negative monomials, and roughly twice the size for the + linear term. +\end_layout + +\begin_layout Section +ABCG version of Ishikawa: +\end_layout + +\begin_layout Standard + +\size footnotesize +\begin_inset Formula +\begin{align} +b_{1}b_{2}\ldots b_{k} & \rightarrow\sum_{i}b_{i}+\sum_{ij}b_{i}b_{j}+\sum_{2i-1}b_{a_{2i-1}}\left(4i-3-\sum_{j}b_{j}\right)\\ +b_{1}b_{2}\ldots b_{k} & \rightarrow\sum_{i}b_{i}+\left(4i-3\right)\sum_{2i-1}b_{a_{2i-1}}+\sum_{ij}b_{i}b_{j}-\sum_{2i-1,j}b_{j}b_{a_{2i-1}} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +Pro. + Same number of auxiliaries as Ishikawa. + Reproduces the full spectrum. +\end_layout + +\begin_layout Standard +Con. + Only works for odd k, but when k is even we can use Ishikawa, so no big + loss. +\end_layout + +\begin_layout Section +Another ABCG version of Ishikawa: +\end_layout + +\begin_layout Standard + +\size footnotesize +\begin_inset Formula +\begin{align} +b_{1}b_{2}\ldots b_{k} & \rightarrow\prod_{i=1}^{k-1}b_{i}-\prod_{i=1}^{k-1}b_{i}(1-b_{k}) +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +Now quadratize the first term using Ishikawa, and use a negative monomial + method for the second term. +\end_layout + +\begin_layout Section +Corollary 4.4 of ABCG +\end_layout + +\begin_layout Section +Corollary 4.5 of ABCG +\end_layout + +\begin_layout Section +Quadratization of +\begin_inset Quotes eld +\end_inset + +parity +\begin_inset Quotes erd +\end_inset + + function on page 17 of ABCG (Theorem 4.6) +\end_layout + +\begin_layout Section +Theorem 5.6 of ABCG +\end_layout + +\begin_layout Section +Theorem 1.1 of Boros-Crama-RodriguezHector (BCR) +\end_layout + +\begin_layout Standard +For any symmetric +\begin_inset Formula $k$ +\end_inset + +-local function that is non-zero (they actually say =1) only if +\begin_inset Formula $\sum b_{i}=m$ +\end_inset + +, if +\begin_inset Formula $\nicefrac{n}{2}\le m\le n$ +\end_inset + + +\end_layout + +\begin_layout Standard + +\size scriptsize +\color black +\begin_inset Formula +\begin{align} +f\left(b_{1},b_{2},\ldots,b_{n}\right) & \rightarrow\left(\sum_{i}b_{i}-\left(m-2^{\lceil\log m\rceil}\right)b_{a_{\lceil\log m\rceil+1}}-(m+1)\left(1-b_{a_{\lceil\log m\rceil+1}}\right)-\sum_{i}^{\lceil\log m\rceil}2^{i-1}b_{a_{i}}\right)^{2}\\ + & =\left(\sum_{i}b_{i}-\left(m-2^{\lceil\log m\rceil}\right)b_{a_{\lceil\log m\rceil+1}}-(m+1)+(m+1)b_{a_{\lceil\log m\rceil+1}}-\sum_{i}^{\lceil\log m\rceil}2^{i-1}b_{a_{i}}\right)^{2}\\ + & =\left(-(m+1)+\sum_{i}b_{i}-\left(2m-2^{\lceil\log m\rceil}+1\right)b_{a_{\lceil\log m\rceil+1}}-\sum_{i}^{\lceil\log m\rceil}2^{i-1}b_{a_{i}}\right)^{2}\\ + & =(m+1)^{2}-2(m+1)\sum_{i}b_{i}+2(m+1)\left(2m-2^{\lceil\log m\rceil}+1\right)b_{a_{\lceil\log m\rceil+1}}+2(m+1)\sum_{i}^{\lceil\log m\rceil}2^{i-1}b_{a_{i}}\\ + & +\sum_{ij}b_{i}b_{j}-2\sum_{i}\left(2m-2^{\lceil\log m\rceil}+1\right)b_{i}b_{a_{\lceil\log m\rceil+1}}-2\sum_{i}\sum_{j}^{\lceil\log m\rceil}2^{i-1}b_{i}b_{a_{j}}+\sum_{i,j}^{\lceil\log m\rceil}2^{i+j-2}b_{a_{i}}b_{a_{j}}\\ + & =\alpha^{I}+\alpha^{b}\sum_{i}b_{i}+\alpha^{b_{a,1}}\sum_{i}^{\lceil\log m\rceil}b_{a_{i}}+\alpha^{b_{a,2}}b_{a_{\lceil\log m\rceil+1}}+\alpha^{bb}\sum_{ij}b_{i}b_{j}+\alpha^{bb_{a,1}}\sum_{i}\sum_{j}^{\lceil\log m\rceil}b_{i}b_{a_{j}}\\ + & +\alpha^{bb_{a,2}}\sum_{i}b_{i}b_{a_{\lceil\log m\rceil+1}}+\alpha^{b_{a}b_{a}}\sum_{i,j}^{\lceil\log m\rceil}b_{a_{i}}b_{a_{j}} +\end{align} + +\end_inset + + +\size default +\color inherit +The number of auxiliary varibales is +\begin_inset Formula $\lceil\log m\rceil+1.$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\begin{pmatrix}\alpha^{I} & \alpha^{bb}\\ +\alpha^{b} & \alpha^{bb_{a,1}}\\ +\alpha^{b_{a,1}} & \alpha^{bb_{a,2}}\\ +\alpha^{b_{a,2}} & \alpha^{b_{a}b_{a}} +\end{pmatrix}=\begin{pmatrix}(m+1)^{2} & 1\\ +-2(m+1) & -2^{i}\\ +2(m+1) & -2\left(2m-2^{\lceil\log m\rceil}+1\right)\\ +2(m+1)\left(2m-2^{\lceil\log m\rceil}+1\right) & 2^{i+j-2} +\end{pmatrix}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Theorem 1.2 of Boros-Crama-RodriguezHector (BCR) +\end_layout + +\begin_layout Standard +For any symmetric +\begin_inset Formula $k$ +\end_inset + +-local function that is non-zero (they actually say =1) only if +\begin_inset Formula $\sum b_{i}=m$ +\end_inset + +, if +\begin_inset Formula $0\le m\le\nicefrac{n}{2}$ +\end_inset + +. +\end_layout + +\begin_layout Standard + +\size scriptsize +\color black +\begin_inset Formula +\begin{align} +f\left(b_{1},b_{2},\ldots,b_{n}\right) & \rightarrow\left(n-\sum_{i}b_{i}-\left(n-m-2^{\lceil\log(n-m)\rceil}\right)b_{a_{\lceil\log(n-m)\rceil+1}}-(n-m+1)\left(1-b_{a_{\lceil\log(n-m)\rceil+1}}\right)-\sum_{i}^{\lceil\log(n-m)\rceil}2^{i-1}b_{a_{i}}\right)^{2}\\ + & =\left(n-\sum_{i}b_{i}-\left(n-m-2^{\lceil\log(n-m)\rceil}\right)b_{a_{\lceil\log(n-m)\rceil+1}}-(n-m+1)+(n-m+1)b_{a_{\lceil\log(n-m)\rceil+1}}-\sum_{i}^{\lceil\log(n-m)\rceil}2^{i-1}b_{a_{i}}\right)^{2}\\ + & =\left(\left(m-1\right)-\sum_{i}b_{i}-\left(2(n-m)-2^{\lceil\log(n-m)\rceil}+1\right)b_{a_{\lceil\log(n-m)\rceil+1}}-\sum_{i}^{\lceil\log(n-m)\rceil}2^{i-1}b_{a_{i}}\right)^{2} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +The number of auxiliary varibales is +\begin_inset Formula $\lceil\log(n-m)\rceil+1.$ +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\begin{pmatrix}\alpha^{I} & \alpha^{bb}\\ +\alpha^{b} & \alpha^{bb_{a,1}}\\ +\alpha^{b_{a,1}} & \alpha^{bb_{a,2}}\\ +\alpha^{b_{a,2}} & \alpha^{b_{a}b_{a}} +\end{pmatrix}=\begin{pmatrix}\\ +\\ +\\ +\\ +\end{pmatrix}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Section +Corollary 1 of BCR +\end_layout + +\begin_layout Standard +transformation not explicitly given, but the function can be more general + than in Theorem 1, but requires a factor of +\begin_inset Formula $\mu$ +\end_inset + + more variables. + +\end_layout + +\begin_layout Section +Theorem 2 of BCR +\end_layout + +\begin_layout Standard +Once again requires typing out a nasty function [should really be done in + mathematica rather than by hand) +\end_layout + +\begin_layout Section +Theorem 4 of BCR +\end_layout + +\begin_layout Standard +This is a special case of Theorem 1, for the specific function +\begin_inset Formula $f=b_{1}b_{2}\ldots b_{k}$ +\end_inset + +. + For som +\begin_inset Formula $p$ +\end_inset + + such that +\begin_inset Formula $k\le2^{p}$ +\end_inset + +, we have: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +b_{1}b_{2}\ldots b_{k} & \rightarrow\left(2^{p}-k+\sum_{i}b_{i}-\sum_{i}2^{i-1}b_{a_{i}}\right)^{2}\\ + & =\left(2^{p}-k\right)^{2}+2\left(2^{p}-k\right)\sum_{i}b_{i}-2\left(2^{p}-k\right)\sum_{i}2^{i-1}b_{a_{i}}+\sum_{ij}b_{i}b_{j}-\sum_{ij}2^{j-1}b_{i}b_{a_{j}}+\sum_{ij}2^{i+-2}b_{a_{i}}b_{a_{j}}\\ + & =\alpha^{I}+\alpha^{b}\sum_{i}b_{i}+\alpha^{b_{a_{i}}}\sum_{i}2^{i-1}b_{a_{i}}+\alpha^{bb}\sum_{ij}b_{i}b_{j}+\alpha^{bb_{a}}\sum_{ij}b_{i}b_{a_{j}}+\alpha^{b_{a_{i}}b_{a_{j}}}b_{a_{i}}b_{a_{j}} +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{equation} +\begin{pmatrix}\alpha^{I} & \alpha^{bb}\\ +\alpha^{b} & \alpha^{bb_{a}}\\ +\alpha^{b_{a}} & \alpha^{b_{a}b_{a}} +\end{pmatrix}=\begin{pmatrix}\left(2^{p}-k\right)^{2} & 1\\ +2\left(2^{p}-k\right) & 2^{j-1}\\ +-2\left(2^{p}-k\right) & 2^{i+-2} +\end{pmatrix}. +\end{equation} + +\end_inset + + +\end_layout + +\begin_layout Standard +Pro: only requires +\begin_inset Formula $\log k$ +\end_inset + + auxiliaries. +\end_layout + +\begin_layout Standard +Con: All terms non-submodular except for the term linear in auxiliaries. +\end_layout + +\begin_layout Section +Theorem 5 of BCR +\end_layout + +\begin_layout Standard +(already written up by Richard) +\end_layout + +\begin_layout Standard +Pro: only requires +\begin_inset Formula $\log k$ +\end_inset + +-1 auxiliaries. +\end_layout + +\begin_layout Section +Theorem 7 of BCR +\end_layout + +\begin_layout Standard +(already written up by Richard) +\end_layout + +\begin_layout Section +Theorem 9 of BCR +\end_layout + +\begin_layout Standard +For the symmetric function which is a function of the sum of all +\begin_inset Formula $n$ +\end_inset + + variables, for some huge integer +\begin_inset Formula $\lambda$ +\end_inset + + such that +\begin_inset Formula $\lambda>\max(f)$ +\end_inset + +, we have: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align} +f\left(\sum b_{i}\right) & \rightarrow\sum_{ij}^{\sqrt{n+1}}f\left((i-1)\left(\lceil\sqrt{n+1}\rceil+1\right)+(j-1)\right)b_{a_{i}}b_{a_{\sqrt{n+1}+j}}+\\ + & +\lambda\left(\left(1-\sum_{i}^{\sqrt{n+1}}b_{a_{i}}\right)^{2}+\left(1-\sum_{i}^{\sqrt{n+1}}b_{a_{\sqrt{n+1}+i}}\right)^{2}+\right.\\ + & +\left.\left(\sum_{i}b_{i}-\left(\left(\lceil\sqrt{n+1}\rceil+1\right)\sum_{i}^{\sqrt{n+1}}(i-1)y_{a_{i}}+\sum_{i}^{\sqrt{n+1}}(i-1)b_{a_{\sqrt{n+1}+i}}\right)\right)^{2}\right) +\end{align} + +\end_inset + + +\end_layout + +\begin_layout Section +Theorem 10 of BCR +\end_layout + +\begin_layout Standard +Works on a generalization of +\begin_inset Formula $f\left(\sum b_{i}\right)$ +\end_inset + + but instead we have a weighted sum. +\end_layout + +\end_body +\end_document diff --git a/MoreReductions.pdf b/MoreReductions.pdf new file mode 100644 index 0000000..06493bd Binary files /dev/null and b/MoreReductions.pdf differ diff --git a/k-local-quadratization.bib b/k-local-quadratization.bib index 8280544..d63cf71 100644 --- a/k-local-quadratization.bib +++ b/k-local-quadratization.bib @@ -555,36 +555,6 @@ @article{Subas2016 volume = {94}, year = {2016} } -@article{Brell2011, -author = {Brell, Courtney G and Flammia, Steven T and Bartlett, Stephen D and Doherty, Andrew C}, -doi = {10.1088/1367-2630/13/5/053039}, -file = {:home/nike/Documents/Mendeley Desktop/Brell et al. - 2011 - Toric codes and quantum doubles from two-body Hamiltonians.pdf:pdf}, -issn = {1367-2630}, -journal = {New Journal of Physics}, -month = {may}, -number = {5}, -pages = {053039}, -publisher = {IOP Publishing}, -title = {{Toric codes and quantum doubles from two-body Hamiltonians}}, -url = {http://stacks.iop.org/1367-2630/13/i=5/a=053039?key=crossref.46bae15eeb0fd6bf104472bfcf5a8fd9}, -volume = {13}, -year = {2011} -} -@article{Brell2014, -author = {Brell, Courtney G and Bartlett, Stephen D and Doherty, Andrew C}, -doi = {10.1088/1367-2630/16/12/123056}, -file = {:home/nike/Documents/Mendeley Desktop/Brell, Bartlett, Doherty - 2014 - Perturbative 2-body parent Hamiltonians for projected entangled pair states.pdf:pdf}, -issn = {1367-2630}, -journal = {New Journal of Physics}, -month = {dec}, -number = {12}, -pages = {123056}, -publisher = {IOP Publishing}, -title = {{Perturbative 2-body parent Hamiltonians for projected entangled pair states}}, -url = {http://stacks.iop.org/1367-2630/16/i=12/a=123056?key=crossref.eeae1f6d6c8ec4881d131c1064f3f52f}, -volume = {16}, -year = {2014} -} @article{Kolmogorov2004, author = {Kolmogorov, V. and Zabih, R.}, doi = {10.1109/TPAMI.2004.1262177}, @@ -735,4 +705,101 @@ @article{Babbush2013 volume = {525}, year = {2013} } - +@article{Shen2017a, +author = {Shen, Jianbing and Peng, Jianteng and Dong, Xingping and Shao, Ling and Porikli, Fatih}, +doi = {10.1109/TIP.2017.2722691}, +issn = {1057-7149}, +journal = {IEEE Transactions on Image Processing}, +month = {oct}, +number = {10}, +pages = {4911--4922}, +title = {{Higher Order Energies for Image Segmentation}}, +url = {http://ieeexplore.ieee.org/document/7967759/}, +volume = {26}, +year = {2017} +} +@article{Ocko2011, +author = {Ocko, Samuel A. and Yoshida, Beni}, +doi = {10.1103/PhysRevLett.107.250502}, +file = {:home/nike/Documents/Mendeley Desktop/Ocko, Yoshida - 2011 - Nonperturbative Gadget for Topological Quantum Codes.pdf:pdf}, +issn = {0031-9007}, +journal = {Physical Review Letters}, +month = {dec}, +number = {25}, +pages = {250502}, +publisher = {American Physical Society}, +title = {{Nonperturbative Gadget for Topological Quantum Codes}}, +url = {https://link.aps.org/doi/10.1103/PhysRevLett.107.250502}, +volume = {107}, +year = {2011} +} +@article{Chancellor2016b, +abstract = {Quantum annealing provides a way of solving optimization problems by encoding them as Ising spin models which are implemented using physical qubits. The solution of the optimization problem then corresponds to the ground state of the system. Quantum tunnelling is harnessed to enable the system to move to the ground state in a potentially highly non-convex energy landscape. A major difficulty in encoding optimization problems in physical quantum annealing devices is the fact that many real world optimization problems require interactions of higher connectivity as well as multi-body terms beyond the limitations of the physical hardware. In this work we address the question of how to implement multi-body interactions using hardware which natively only provides two-body interactions. The main result is an efficient circuit design of such multi-body terms using superconducting flux qubits in which effective N-body interactions are implemented using N ancilla qubits and only two inductive couplers. It is then shown how this circuit can be used as the unit cell of a scalable architecture by applying it to a recently proposed embedding technique for constructing an architecture of logical qubits with arbitrary connectivity using physical qubits which have nearest-neighbor four-body interactions. It is further shown that this design is robust to non-linear effects in the coupling loops as well as mismatches in some of the circuit parameters.}, +archivePrefix = {arXiv}, +arxivId = {1603.09521}, +author = {Chancellor, Nicholas and Zohren, Stefan and Warburton, Paul A.}, +doi = {10.1038/s41534-017-0022-6}, +eprint = {1603.09521}, +file = {::}, +issn = {2056-6387}, +month = {mar}, +title = {{Circuit design for multi-body interactions in superconducting quantum annealing system with applications to a scalable architecture}}, +url = {http://arxiv.org/abs/1603.09521{\%}0Ahttp://dx.doi.org/10.1038/s41534-017-0022-6}, +year = {2016} +} +@article{Leib2016a, +abstract = {Adiabatic quantum computing is an analog quantum computing scheme with various applications in solving optimization problems. In the parity picture of quantum optimization, the problem is encoded in local fields that act on qubits which are connected via local 4-body terms. We present an implementation of a parity annealer with Transmon qubits with a specifically tailored Ising interaction from Josephson ring modulators.}, +archivePrefix = {arXiv}, +arxivId = {1604.02359}, +author = {Leib, Martin and Zoller, Peter and Lechner, Wolfgang}, +eprint = {1604.02359}, +file = {::}, +month = {apr}, +title = {{A Transmon Quantum Annealer: Decomposing Many-Body Ising Constraints Into Pair Interactions}}, +url = {http://arxiv.org/abs/1604.02359}, +year = {2016} +} +@article{Kahl2011, +author = {Kahl and Strandmark}, +journal = {International Conference on Computer Vision}, +mendeley-groups = {csp-reduce,Review Paper}, +title = {{Generalized Roof Duality for Pseudo-Boolean Optimization}}, +url = {http://www.maths.lth.se/vision/publdb/reports/pdf/kahl-strandmark-iccv-11.pdf}, +year = {2011} +} +@inproceedings{Boros2018QuadratizationsOS, + title={Quadratizations of symmetric pseudo-Boolean functions: sub-linear bounds on the number of auxiliary variables}, + author={Endre Boros and Yves Crama and Elisabeth Rodr{\'i}guez-Heck}, + booktitle={ISAIM}, + year={2018} +} +@article{Brell2011, +author = {Brell, Courtney G and Flammia, Steven T and Bartlett, Stephen D and Doherty, Andrew C}, +doi = {10.1088/1367-2630/13/5/053039}, +file = {:home/nike/Documents/Mendeley Desktop/Brell et al. - 2011 - Toric codes and quantum doubles from two-body Hamiltonians.pdf:pdf}, +issn = {1367-2630}, +journal = {New Journal of Physics}, +month = {may}, +number = {5}, +pages = {053039}, +publisher = {IOP Publishing}, +title = {{Toric codes and quantum doubles from two-body Hamiltonians}}, +url = {http://stacks.iop.org/1367-2630/13/i=5/a=053039?key=crossref.46bae15eeb0fd6bf104472bfcf5a8fd9}, +volume = {13}, +year = {2011} +} +@article{Brell2014, +author = {Brell, Courtney G and Bartlett, Stephen D and Doherty, Andrew C}, +doi = {10.1088/1367-2630/16/12/123056}, +file = {:home/nike/Documents/Mendeley Desktop/Brell, Bartlett, Doherty - 2014 - Perturbative 2-body parent Hamiltonians for projected entangled pair states.pdf:pdf}, +issn = {1367-2630}, +journal = {New Journal of Physics}, +month = {dec}, +number = {12}, +pages = {123056}, +publisher = {IOP Publishing}, +title = {{Perturbative 2-body parent Hamiltonians for projected entangled pair states}}, +url = {http://stacks.iop.org/1367-2630/16/i=12/a=123056?key=crossref.eeae1f6d6c8ec4881d131c1064f3f52f}, +volume = {16}, +year = {2014} +} \ No newline at end of file diff --git a/k-local-review.bbl b/k-local-review.bbl index 957ec4c..bb94534 100644 --- a/k-local-review.bbl +++ b/k-local-review.bbl @@ -6,7 +6,7 @@ %Control: page (0) single %Control: year (1) truncated %Control: production of eprint (0) enabled -\begin{thebibliography}{35}% +\begin{thebibliography}{41}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} @@ -89,6 +89,15 @@ }\href {http://arxiv.org/abs/1508.07190} {\ ,\ \bibinfo {pages} {5} (\bibinfo {year} {2015})},\ \Eprint {http://arxiv.org/abs/1508.07190} {arXiv:1508.07190} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Kolmogorov}\ and\ \citenamefont + {Zabih}(2004)}]{Kolmogorov2004}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont + {Kolmogorov}}\ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont + {Zabih}},\ }\href {\doibase 10.1109/TPAMI.2004.1262177} {\bibfield {journal} + {\bibinfo {journal} {IEEE Transactions on Pattern Analysis and Machine + Intelligence}\ }\textbf {\bibinfo {volume} {26}},\ \bibinfo {pages} {147} + (\bibinfo {year} {2004})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Freedman}\ and\ \citenamefont {Drineas}(2005)}]{Freedman2005}% \BibitemOpen @@ -124,15 +133,37 @@ }\href {http://arxiv.org/abs/1404.6538} {\ (\bibinfo {year} {2014})},\ \Eprint {http://arxiv.org/abs/1404.6538} {arXiv:1404.6538} \BibitemShut {NoStop}% -\bibitem [{\citenamefont {Kolmogorov}\ and\ \citenamefont - {Zabih}(2004)}]{Kolmogorov2004}% +\bibitem [{\citenamefont {Chancellor}\ \emph + {et~al.}(2016{\natexlab{a}})\citenamefont {Chancellor}, \citenamefont + {Zohren},\ and\ \citenamefont {Warburton}}]{Chancellor2016b}% \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont - {Kolmogorov}}\ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont - {Zabih}},\ }\href {\doibase 10.1109/TPAMI.2004.1262177} {\bibfield {journal} - {\bibinfo {journal} {IEEE Transactions on Pattern Analysis and Machine - Intelligence}\ }\textbf {\bibinfo {volume} {26}},\ \bibinfo {pages} {147} - (\bibinfo {year} {2004})}\BibitemShut {NoStop}% + \bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont + {Chancellor}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Zohren}}, + \ and\ \bibinfo {author} {\bibfnamefont {P.~A.}\ \bibnamefont {Warburton}},\ + }\href {\doibase 10.1038/s41534-017-0022-6} {\ (\bibinfo {year} + {2016}{\natexlab{a}}),\ 10.1038/s41534-017-0022-6},\ \Eprint + {http://arxiv.org/abs/1603.09521} {arXiv:1603.09521} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Leib}\ \emph + {et~al.}(2016{\natexlab{a}})\citenamefont {Leib}, \citenamefont {Zoller},\ + and\ \citenamefont {Lechner}}]{Leib2016a}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Leib}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Zoller}}, \ and\ + \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Lechner}},\ }\href + {http://arxiv.org/abs/1604.02359} {\ (\bibinfo {year} + {2016}{\natexlab{a}})},\ \Eprint {http://arxiv.org/abs/1604.02359} + {arXiv:1604.02359} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Leib}\ \emph + {et~al.}(2016{\natexlab{b}})\citenamefont {Leib}, \citenamefont {Zoller},\ + and\ \citenamefont {Lechner}}]{Leib2016}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Leib}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Zoller}}, \ and\ + \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Lechner}},\ }\href + {http://stacks.iop.org/2058-9565/1/i=1/a=015008} {\bibfield {journal} + {\bibinfo {journal} {Quantum Science and Technology}\ }\textbf {\bibinfo + {volume} {1}},\ \bibinfo {pages} {15008} (\bibinfo {year} + {2016}{\natexlab{b}})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Gallagher}\ \emph {et~al.}(2011)\citenamefont {Gallagher}, \citenamefont {Batra},\ and\ \citenamefont {Parikh}}]{Gallagher2011}% @@ -161,9 +192,10 @@ System}},}\ }\bibinfo {howpublished} {AQC 2016 https://www.youtube.com/watch?v=aC-6hg{\_}h3EA} (\bibinfo {year} {2016})\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Chancellor}\ \emph {et~al.}(2016)\citenamefont - {Chancellor}, \citenamefont {Zohren}, \citenamefont {Warburton}, - \citenamefont {Benjamin},\ and\ \citenamefont {Roberts}}]{Chancellor2016}% +\bibitem [{\citenamefont {Chancellor}\ \emph + {et~al.}(2016{\natexlab{b}})\citenamefont {Chancellor}, \citenamefont + {Zohren}, \citenamefont {Warburton}, \citenamefont {Benjamin},\ and\ + \citenamefont {Roberts}}]{Chancellor2016}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Chancellor}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Zohren}}, @@ -171,8 +203,25 @@ {author} {\bibfnamefont {S.}~\bibnamefont {Benjamin}}, \ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Roberts}},\ }\href {\doibase 10.1038/srep37107} {\bibfield {journal} {\bibinfo {journal} {Scientific - Reports}\ }\textbf {\bibinfo {volume} {6}} (\bibinfo {year} {2016}),\ - 10.1038/srep37107}\BibitemShut {NoStop}% + Reports}\ }\textbf {\bibinfo {volume} {6}} (\bibinfo {year} + {2016}{\natexlab{b}}),\ 10.1038/srep37107}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Kahl}\ and\ \citenamefont + {Strandmark}(2011)}]{Kahl2011}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibnamefont {Kahl}}\ and\ \bibinfo + {author} {\bibnamefont {Strandmark}},\ }\href + {http://www.maths.lth.se/vision/publdb/reports/pdf/kahl-strandmark-iccv-11.pdf} + {\bibfield {journal} {\bibinfo {journal} {International Conference on + Computer Vision}\ } (\bibinfo {year} {2011})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Boros}\ \emph {et~al.}(2018)\citenamefont {Boros}, + \citenamefont {Crama},\ and\ \citenamefont + {Rodr{\'i}guez-Heck}}]{Boros2018QuadratizationsOS}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont + {Boros}}, \bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont {Crama}}, \ and\ + \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Rodr{\'i}guez-Heck}},\ + }in\ \href@noop {} {\emph {\bibinfo {booktitle} {ISAIM}}}\ (\bibinfo {year} + {2018})\BibitemShut {NoStop}% \bibitem [{\citenamefont {Rosenberg}(1975)}]{Rosenberg1975}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {I.~G.}\ \bibnamefont @@ -230,16 +279,31 @@ {Cubitt}},\ }\href {\doibase 10.1126/science.aab3326} {\bibfield {journal} {\bibinfo {journal} {Science}\ }\textbf {\bibinfo {volume} {351}},\ \bibinfo {pages} {1180} (\bibinfo {year} {2016})}\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Leib}\ \emph {et~al.}(2016)\citenamefont {Leib}, - \citenamefont {Zoller},\ and\ \citenamefont {Lechner}}]{Leib2016}% +\bibitem [{\citenamefont {Ocko}\ and\ \citenamefont + {Yoshida}(2011)}]{Ocko2011}% \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont - {Leib}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Zoller}}, \ and\ - \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Lechner}},\ }\href - {http://stacks.iop.org/2058-9565/1/i=1/a=015008} {\bibfield {journal} - {\bibinfo {journal} {Quantum Science and Technology}\ }\textbf {\bibinfo - {volume} {1}},\ \bibinfo {pages} {15008} (\bibinfo {year} + \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~A.}\ \bibnamefont + {Ocko}}\ and\ \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Yoshida}},\ + }\href {\doibase 10.1103/PhysRevLett.107.250502} {\bibfield {journal} + {\bibinfo {journal} {Physical Review Letters}\ }\textbf {\bibinfo {volume} + {107}},\ \bibinfo {pages} {250502} (\bibinfo {year} {2011})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Subasi}\ and\ \citenamefont + {Jarzynski}(2016)}]{Subas2016}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont + {Subasi}}\ and\ \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont + {Jarzynski}},\ }\href {\doibase 10.1103/PhysRevA.94.012342} {\bibfield + {journal} {\bibinfo {journal} {Physical Review A}\ }\textbf {\bibinfo + {volume} {94}},\ \bibinfo {pages} {012342} (\bibinfo {year} {2016})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Nagaj}(2010)}]{Nagaj2010}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont + {Nagaj}},\ }\href {\doibase 10.1063/1.3384661} {\bibfield {journal} + {\bibinfo {journal} {Journal of Mathematical Physics}\ }\textbf {\bibinfo + {volume} {51}},\ \bibinfo {pages} {062201} (\bibinfo {year} + {2010})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Duan}\ and\ \citenamefont {Chen}(2011)}]{Duan2011}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Q.-H.}\ \bibnamefont @@ -298,22 +362,6 @@ {\doibase 10.1088/1367-2630/13/5/053039} {\bibfield {journal} {\bibinfo {journal} {New Journal of Physics}\ }\textbf {\bibinfo {volume} {13}},\ \bibinfo {pages} {053039} (\bibinfo {year} {2011})}\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Nagaj}(2010)}]{Nagaj2010}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont - {Nagaj}},\ }\href {\doibase 10.1063/1.3384661} {\bibfield {journal} - {\bibinfo {journal} {Journal of Mathematical Physics}\ }\textbf {\bibinfo - {volume} {51}},\ \bibinfo {pages} {062201} (\bibinfo {year} - {2010})}\BibitemShut {NoStop}% -\bibitem [{\citenamefont {Subasi}\ and\ \citenamefont - {Jarzynski}(2016)}]{Subas2016}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont - {Subasi}}\ and\ \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont - {Jarzynski}},\ }\href {\doibase 10.1103/PhysRevA.94.012342} {\bibfield - {journal} {\bibinfo {journal} {Physical Review A}\ }\textbf {\bibinfo - {volume} {94}},\ \bibinfo {pages} {012342} (\bibinfo {year} - {2016})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Babbush}\ \emph {et~al.}(2013)\citenamefont {Babbush}, \citenamefont {O'Gorman},\ and\ \citenamefont {Aspuru-Guzik}}]{Babbush2013}% @@ -370,4 +418,16 @@ {Li}},\ }\href {\doibase 10.1126/sciadv.1601246} {\bibfield {journal} {\bibinfo {journal} {Science Advances}\ }\textbf {\bibinfo {volume} {2}} (\bibinfo {year} {2016}),\ 10.1126/sciadv.1601246}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Shen}\ \emph {et~al.}(2017)\citenamefont {Shen}, + \citenamefont {Peng}, \citenamefont {Dong}, \citenamefont {Shao},\ and\ + \citenamefont {Porikli}}]{Shen2017a}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Shen}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Peng}}, \bibinfo + {author} {\bibfnamefont {X.}~\bibnamefont {Dong}}, \bibinfo {author} + {\bibfnamefont {L.}~\bibnamefont {Shao}}, \ and\ \bibinfo {author} + {\bibfnamefont {F.}~\bibnamefont {Porikli}},\ }\href {\doibase + 10.1109/TIP.2017.2722691} {\bibfield {journal} {\bibinfo {journal} {IEEE + Transactions on Image Processing}\ }\textbf {\bibinfo {volume} {26}},\ + \bibinfo {pages} {4911} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% \end{thebibliography}% diff --git a/k-local-review.pdf b/k-local-review.pdf index c18e413..38c1433 100644 Binary files a/k-local-review.pdf and b/k-local-review.pdf differ diff --git a/k-local-review.tex b/k-local-review.tex index caf7b43..c15082e 100644 --- a/k-local-review.tex +++ b/k-local-review.tex @@ -127,7 +127,7 @@ %\email{jacob.biamonte@qubit.org} -\author{Nikesh S.~Dattani} +\author{Nike Dattani} \email{nik.dattani@gmail.com} \affiliation{Oxford University, Department of Mathematics} \affiliation{McMaster University, Department of Chemistry and Chemical Biology} @@ -188,15 +188,17 @@ \vspace{-5mm} \begin{align} -s_1s_2+s_2s_3+s_3s_4 -4s_1s_2s_3 & \hspace{5mm} \textrm{(cubic)},\label{eq:intro_quantum_cubic} +z_1x_2+x_2y_3+z_3z_4 -4y_1y_2z_3 & \hspace{5mm} \textrm{(cubic)},\label{eq:intro_quantum_cubic} \end{align} -\noindent where each spin $s_i$ can be any of the Pauli matrices, and the coefficients tell us about the strengths of couplings between these particles. The Schr\"{o}dinger equation tells us that the eigenvalues of the Hamiltonian are the allowed energy levels and their eigenvectors (wavefunctions) are the corresponding physical states. More generally these do not have to be spins but can be any type of qubits, and we can encode the solution to \textit{any} problem in the ground state of a Hamiltonian, then solve the problem by finding the lowest energy state of the physical system (this is called adiabatic quantum computing). +%where each spin $s_i$ can be any of the Pauli matrices, and + +\noindent where the coefficients tell us about the strengths of couplings between these particles. The Schr\"{o}dinger equation tells us that the eigenvalues of the Hamiltonian are the allowed energy levels and their eigenvectors (wavefunctions) are the corresponding physical states. More generally these do not have to be spins but can be any type of qubits, and we can encode the solution to \textit{any} problem in the ground state of a Hamiltonian, then solve the problem by finding the lowest energy state of the physical system (this is called adiabatic quantum computing). Two-body physical interactions occur more naturally than many-body interactions so \textbf{Parts \ref{partTransverseIsing}-\ref{partGeneral}} give more than 10 different ways to quadratize genreal Hamiltonians (some of these methods do not even require the $s_i$ to be $2\times2$ matrices, meaning that we can have types of qudits that are not qubits). %A quantum computer whose Hamiltonian is at least quadratic in $x$ and $z$ can find this ground state with only polynomial runtime overhead over the best alternative algorithm to find that solution. - The optimization problems of Eqs. \eqref{eq:intro_cubic}-\eqref{eq:intro_quadratic} are specific cases of Eq. \eqref{eq:intro_quantum_cubic} where only $z$ matrices are present. + The optimization problems of Eqs. \eqref{eq:intro_cubic}-\eqref{eq:intro_quadratic} are specific cases of the type in Eq. \eqref{eq:intro_quantum_cubic} where only $z$ matrices are present. \noindent\rule{\textwidth}{0.4pt} @@ -270,7 +272,7 @@ \subsection{Deduction Reduction (Deduc-reduc)} \label{subsec:deduc_reduc} \costsec \begin{itemize} \item $0$ auxiliary variables needed.% Given a deduction, the reduction time is proportional to the number of terms in the Hamiltonian and is negligible. (but this is true of all the other reducs too) -\item To find every possible $m$-variable deduction for $n$ total variables would involve $3^n - 2^n -1$ evaluations of the objective function, whereas $2^n$ evaluations is enough to solve the entire problem. We therefore choose $m \lll n$. +\item For a particular value of $m$, we have $\binom{n}{m}$ different $m$-variable subsets of the $n$ variable problem, and $\binom{n}{m} 2^m$ evaluations of the objective function to find all possible $m$-variable deductions, whereas $2^n$ evaluations is enough to solve the entire problem. We therefore choose $m \lll n$. %The computational cost of the search for deductions is difficult to estimate. %The approximate worst-case complexity is $\mathcal{O}(n^{d+1}2^{m})$ where $m$ is the number of variables in a 'small' problem, $n$ is the total number of variables and $d$ is the maximum degree of deductions we are searching for. %We suggest $10 \le m \le 20$, so that a small problem involves checking roughly $1,000$ to $1,000,000$ states, and $d=2$. @@ -330,7 +332,8 @@ \subsection{ELC Reduction} \costsec \begin{itemize} \item $0$ auxiliary variables needed.% Given a deduction, the reduction time is proportional to the number of terms in the Hamiltonian and is negligible. (but this is true of all the other reducs too) -\item To find every possible $m$-variable deduction for $n$ total variables would involve $3^n - 2^n -1$ evaluations of the objective function, whereas $2^n$ evaluations is enough to solve the entire problem. We therefore choose $m \lll n$. +\item For a particular value of $m$, we have $\binom{n}{m}$ different $m$-variable subsets of the $n$ variable problem, and $\binom{n}{m} 2^m$ evaluations of the objective function to find all possible $m$-variable deductions, whereas $2^n$ evaluations is enough to solve the entire problem. We therefore choose $m \lll n$. +%\item To find every possible $m$-variable deduction for $n$ total variables would involve $3^n - 2^n -1$ evaluations of the objective function, whereas $2^n$ evaluations is enough to solve the entire problem. We therefore choose $m \lll n$. %The computational cost of the search for deductions is difficult to estimate. %The approximate worst-case complexity is $\mathcal{O}(n^{d+1}2^{m})$ where $m$ is the number of variables in a 'small' problem, $n$ is the total number of variables and $d$ is the maximum degree of deductions we are searching for. %We suggest $10 \le m \le 20$, so that a small problem involves checking roughly $1,000$ to $1,000,000$ states, and $d=2$. @@ -455,7 +458,7 @@ \subsection{Groebner Bases} \conssec \begin{itemize} \item Best algorithms for finding Groebner bases scale double exponentially in $n$. -\item Only works for Hamiltonians whose minimization corresponds to solving systems of discrete equations (RICHARD, why is this the ONLY case?). +\item Only works for Hamiltonians whose minimization corresponds to solving systems of discrete equations, as the method only preserves roots, not minima. \end{itemize} \examplesec @@ -563,7 +566,8 @@ \subsection{Negative Term Reduction} \label{subsec:Negative-Monomial-Reduction} For a negative term $-b_{1}b_{2}...b_{k}$, introduce a single auxiliary variable $b_a$ and make the substitution: \begin{equation} --b_{1}b_{2} \ldots b_{k} = \min_{b_a} \left( (k-1-\sum_i b_{i})b_a \right). +%-b_{1}b_{2} \ldots b_{k} = \min_{b_a} \left( (k-1-\sum_i b_{i})b_a \right). +-b_{1}b_{2} \ldots b_{k} \rightarrow \left( (k-1-\sum_i b_{i})b_a \right). \end{equation} \costsec @@ -575,6 +579,7 @@ \subsection{Negative Term Reduction} \label{subsec:Negative-Monomial-Reduction} \begin{itemize} \item All resulting quadratic terms are submodular (have negative coefficients). \item Can reduce arbitrary order terms with only 1 auxiliary. +\item Reproduces the full spectrum. %\item Can be generalized so that one variable can be made to work for multiple terms.% (I can't find anyone saying this, but can be proven. Look at the commented out example below!) \end{itemize} @@ -611,12 +616,15 @@ \subsection{Negative Term Reduction} \label{subsec:Negative-Monomial-Reduction} \refsec \begin{itemize} -\item Original paper: \cite{Freedman2005}. +\item 2004: Kolmogorov and Zabih presented this for cubic terms \cite{Kolmogorov2004}. +\item 2005: Generalized to arbitrary order by Freedman and Drineas \cite{Freedman2005}. \item Discussion: \cite{Ishikawa2011}, \cite{Anthony2015}. \end{itemize} \newpage + + \subsection{Positive Term Reduction} \summarysec @@ -624,7 +632,8 @@ \subsection{Positive Term Reduction} By considering the negated literals $\bar{b}_{i}=1-b_{i}$, we recursively apply the previous method to $b_{1}b_{2}\ldots b_{k}=-\bar{b}_{1}b_{2}\ldots b_{k}+b_{2}b_{3}\ldots b_{k}$. The final identity is: \begin{equation} -b_{1}b_{2}\ldots b_{k}=\min_{b_a}\left(\sum_{i=1}^{k-2}b_{a_{i}}(k-i-1+b_{i}-\sum_{j=i+1}^{k}b_{j})\right)+b_{k-1}b_{k} +b_{1}b_{2}\ldots b_{k}\rightarrow \left(\sum_{i=1}^{k-2}b_{a_{i}}(k-i-1+b_{i}-\sum_{j=i+1}^{k}b_{j})\right)+b_{k-1}b_{k} +%b_{1}b_{2}\ldots b_{k}=\min_{b_a}\left(\sum_{i=1}^{k-2}b_{a_{i}}(k-i-1+b_{i}-\sum_{j=i+1}^{k}b_{j})\right)+b_{k-1}b_{k} \end{equation} @@ -645,7 +654,8 @@ \subsection{Positive Term Reduction} \examplesec \begin{eqnarray} -b_{1}b_{2}b_{3}b_{4} & = & \min_{b_a}{b_{a_{1}}(2+b_{1}-b_{2}-b_{3}-b_{4})+b_{a_{2}}(1+b_{2}-b_{3}-b_{4})}+b_{3}b_{4} +%b_{1}b_{2}b_{3}b_{4} & = & \min_{b_a}{b_{a_{1}}(2+b_{1}-b_{2}-b_{3}-b_{4})+b_{a_{2}}(1+b_{2}-b_{3}-b_{4})}+b_{3}b_{4} +b_{1}b_{2}b_{3}b_{4} & \rightarrow {b_{a_{1}}(2+b_{1}-b_{2}-b_{3}-b_{4})+b_{a_{2}}(1+b_{2}-b_{3}-b_{4})}+b_{3}b_{4} \end{eqnarray} \refsec @@ -671,9 +681,10 @@ \subsection{Ishikawa's Symmetric Reduction (Positive Term Reduction)} \summarysec -This method rewrites a positive monomial using symmetric polynomials, so all possible quadratic terms are produced and they are all non-submodular: +This method re-writes a positive monomial using symmetric polynomials, so all possible quadratic terms are produced and they are all non-submodular: \begin{equation} -b_{1}...b_{k} = \min_{b_{a_1}, \ldots, b_{a_{n_k}}} \left( \sum_{i=1}^{n_{k}}b_{a_{i}}\left(c_{i,d}\left(-\sum_{j=1}^{k}b_{j}+2i\right)-1\right)+\sum_{i 2$. +Let $X = \sum^k_{i=1} b_i$. + +Let $m = \lceil \frac{k}{4} \rceil, N = n - 2m$ and +$Y = \sum^m_{j=2} b'_j$ where $b'_1, \ldots, b'_m$ are auxiliary variables. +Note that the $j$ index starts at 2. +Then: + +\begin{align} +b_1 \ldots b_k &= \min_{b'_1, \ldots b'_m} \frac{1}{2} \left( X - Nb'_1 - 2Y \right) +\left( X - Nb'_1 - 2Y - 1 \right) +\end{align} + +\costsec + +$\lceil \frac{k}{4} \rceil$ auxiliary qubits per positive monomial. + +\prossec +\begin{itemize} +\item Smallest number of auxiliary coefficients that scales linearly with $k$. +\item Smaller coefficients than the logarithmic reduction. +\end{itemize} + +\conssec +\begin{itemize} +\item Introduces many non-submodular terms. +\end{itemize} + +\examplesec + +Consider $k=4$, so that $m=1$, $N = 2$. +Then: +\begin{eqnarray} +b_1 b_2 b_3 b_4 = \min_{b'} \frac{1}{2} +\left( b_1 + b_2 + b_3 + b_4 - 2b'_1 \right) +\left( b_1 + b_2 + b_3 + b_4 - 2b'_1 - 1 \right) +\end{eqnarray} + +\refsec +\begin{itemize} +\item Original paper: \cite{Boros2018QuadratizationsOS}. +\end{itemize} + +\newpage + +\subsection{Logarithmic Positive Monomial Reduction} \summarysec -This method can also be used to rewrite positive cubic terms in terms of 6 quadratic terms. +Suppose we have a monomial $b_1b_2b_3 \ldots b_k$. +Pick $n$ minimal such that $k < 2^{n+1}$ and let $N = 2^{n+1} - k$. + +For convenience, let $X = \sum^k_{i=1} b_i$. + +\begin{align} +b_1 \ldots b_k &= \min_{b'_1, \ldots b'_n} \frac{1}{2} \left( N + X - \sum^n_{i=1} 2^i b'_i \right) +\left( N + X - \sum^n_{i=1} 2^i b'_i - 1 \right) +\end{align} + +\costsec + +$\lceil \log k \rceil - 1$ auxiliary qubits per positive monomial. + +\prossec +\begin{itemize} +\item Logarithmic number of auxiliary variables. +\end{itemize} + +\conssec +\begin{itemize} +\item Introduces many non-submodular terms. +\end{itemize} + +\examplesec + +Consider $k=4$, so that $n=2$, $N = 4$. +Then: +\begin{eqnarray} +b_1 b_2 b_3 b_4 = \min_{b'_1, b'_2} \frac{1}{2} +\left( 4 + b_1 + b_2 + b_3 + b_4 - b'_1 - 2b'_2 \right) +\left( 3 + b_1 + b_2 + b_3 + b_4 - b'_1 - 2b'_2 \right) +\end{eqnarray} + +\refsec +\begin{itemize} +\item Original paper: \cite{Boros2018QuadratizationsOS}. +\end{itemize} + +\newpage + +\subsection{Reduction by Minimum Selection (Kolmogorov \& Zabih, 2004)} + +\summarysec +This method can be used to re-write positive or negative cubic terms in terms of 6 quadratic terms. The identity is given by: \begin{align} -b_{1}b_{2}b_{3} &= \min_{b_a} - \left( b_a + b_1 + b_2 + b_3 \right) + b_a \left( b_1 + b_2 + b_3 \right) + b_1 b_2 + b_2 b_3 + b_3 b_1 +b_{1}b_{2}b_{3} & \rightarrow 1 - \left( b_a + b_1 + b_2 + b_3 \right) + b_a \left( b_1 + b_2 + b_3 \right) + b_1 b_2 + b_1 b_3 + b_2 b_3 \end{align} \costsec +\begin{itemize} +\item 1 auxiliary variable per positive or negative cubic term. +\end{itemize} -1 auxiliary variable per positive cubic term. +\prossec +\begin{itemize} +\item Works on positive or negative monomials. +\item Reproduces the full spectrum. +\end{itemize} + +\conssec +\begin{itemize} +\item Introduces all 6 possible non-submodular quadratic terms. +\end{itemize} + +\refsec +\begin{itemize} +\item Original paper: \cite{Kolmogorov2004}. +\end{itemize} + +\newpage + + +\subsection{Reduction by Minimum Selection (in terms of $z$)} + +\summarysec + +The formula is almost the same as in the version in terms of $b$, but with a factor of 2, a change of sign for the linear terms, and a slight change in the constant term: + +\begin{align} +\pm z_1z_2z_3 &= 3 \pm \left( z_1 + z_2 + z_3 + z_a \right) + 2z_a \left(z_1 + z_2 +z_3 \right) + z_1z_2 + z_1z_3 + z_2z_3. +\end{align} + +\costsec +\begin{itemize} +\item 1 auxiliary variable per positive or negative cubic term. +\end{itemize} \prossec \begin{itemize} -\item Works on positive monomials. +\item Works on positive or negative monomials. +\item Reproduces the full spectrum. \end{itemize} \conssec \begin{itemize} -\item Introduces all 6 possible non-submodular terms. +\item Introduces all 6 possible non-submodular quadratic terms. \end{itemize} \refsec \begin{itemize} -\item Original introduction by Kolmogorov and Zabih: \cite{Kolmogorov2004}. +\item 31 March 2016 published by Chancellor, Zohren, and Warburton without the constant term: \cite{Chancellor2016b}. +\item 8 April 2016 published independently by Leib, Zoller, and Lechner without the constant term \cite{Leib2016a,Leib2016}. \end{itemize} \newpage @@ -753,7 +901,7 @@ \subsection{Asymmetric Reduction} \begin{align} b_{1}b_{2}b_{3}&= \min_{b_a} \left( b_a-b_{2}b_a-b_{3}b_a+b_{1}b_a+b_{2}b_{3} \right)\\ &= \min_{b_a} \left( b_a-b_{1}b_a-b_{3}b_a+b_{2}b_a+b_{1}b_{3} \right)\\ - &= \min_{b_a} \left( b_a-b_{1}b_a-b_{2}b_a+b_{3}b_a+b_{1}b_{2} \right)\\ + &= \min_{b_a} \left( b_a-b_{1}b_a-b_{2}b_a+b_{3}b_a+b_{1}b_{2} \right).\\ \end{align} \costsec @@ -771,6 +919,11 @@ \subsection{Asymmetric Reduction} \item Only been shown to work for cubics. \end{itemize} +\examplesec +\begin{eqnarray} +b_{1}b_{2}b_{3} + b_1 b_3 - b_2 &= \min_{b_a} \left( b_a-b_{1}b_a-b_{3}b_a+b_{2}b_a+2b_{1}b_{3} \right) - b_2 +\end{eqnarray} + \refsec \begin{itemize} \item Original paper and application to computer vision: \cite{Gallagher2011}. @@ -827,6 +980,7 @@ \subsection{Bit flipping} \item Original paper: \cite{Ishikawa2011}. \end{itemize} +\newpage \subsection{Symmetry Based Mappings} \summarysec @@ -897,6 +1051,40 @@ \subsection{Symmetry Based Mappings} \item Talk including use in quantum simulation: \cite{Chancellor2016a} \end{itemize} +\subsection{Another Cubic Reduction} + +\summarysec +\begin{align} +-b_{1}b_{2}b_{3}&= \min_{b_a} b_a \left( -b_1 + b_2 + b_3 \right) -b_1b_2 - b_1b_3 + b_1 +\end{align} + +\costsec + +1 auxiliary variable per negative cubic term. + +\prossec +\begin{itemize} +\item Asymmetric which allows more flexibility in cancelling with other quadratics. +\end{itemize} + +\conssec +\begin{itemize} +\item Only works for negative cubic monomials. +\end{itemize} + +\examplesec +\begin{eqnarray} +- b_{1}b_{2}b_{3} + b_1 b_3 - b_2 &= \min_{b_a} \left( b_a-b_{1}b_a-b_{3}b_a+b_{2}b_a+2b_{1}b_{3} \right) - b_2 +\end{eqnarray} + +\refsec +\begin{itemize} +\item Discussion: \cite{Ishikawa2011}, \cite{Kahl2011}. +\end{itemize} + +\newpage + + \section{Methods that introduce auxiliary variables to quadratize MULTIPLE terms with the SAME auxiliaries} \subsection{Reduction by Substitution} @@ -904,7 +1092,7 @@ \subsection{Reduction by Substitution} \summarysec Pick a variable pair $(b_{i},b_{j})$ and substitute $b_{i}b_{j}$ with a new auxiliary variable $b_{a_{ij}}$. -Enforce equality in the ground states by adding some scalar multiple of the penalty $P=b_{i}b_{j}-2b_{i}b_{a_{ij}}-2b_{i}b_{a_{ij}}+3b_{a_{ij}}$ or similar. Since P > 0 if and only if $b_{a_{ij}}\ne b_ib_j$, the minimum of the new $(k-1)$-local function will satisfy $b_{a_ij}=b_{i}b_{j})$, which means that at the minimum, we have precisely the original function. Repeat $(k-2)$ times for each $k$-local term and the resulting function will be 2-local. +Enforce equality in the ground states by adding some scalar multiple of the penalty $P=b_{i}b_{j}-2b_{i}b_{a_{ij}}-2b_{i}b_{a_{ij}}+3b_{a_{ij}}$ or similar. Since $P > 0$ if and only if $b_{a_{ij}}\ne b_ib_j$, the minimum of the new $(k-1)$-local function will satisfy $b_{a_ij}=b_{i}b_{j})$, which means that at the minimum, we have precisely the original function. Repeat $(k-2)$ times for each $k$-local term and the resulting function will be 2-local. \costsec \begin{itemize} @@ -1156,13 +1344,14 @@ \subsection{Flag Based SAT Mapping\label{sub:flag_SAT}} \newpage + \subsection{Chained Three Body Parity Operators} \summarysec -Goal: Guarantee that $z_{a_{k}}=z_iz_j$ can be chained together to make large product terms consisting of $z$. +Goal: Terms of the form $z_{a_{k}}=-z_iz_j$ can be chained together to make large product terms consisting of $z$. %Recall that products of $b$ correspond to parity check or ${\sc xor}$ operations, while products of $b$ correspond to logical ${\sc and}$ operations. -The penalty term $P(z_{a_k},z_i,z_j)= \mp z_{a_k}z_iz_j$ guarantees that $z_{a_k}=\pm z_iz_j$. The 3-local term can be made from gadgets. +The penalty term $P(z_{a_k},z_i,z_j)= \mp z_{a_k}z_iz_j$ will lead to an energy penalty unless $z_{a_k}=\pm z_iz_j$. The 3-local term can be made from gadgets. %This method was originally only used to reproduce the ground state of high locality terms, but states of the "wrong" parity ($q_k=\mp z_iz_j$) will all have the same energy as well, so it reproduces the full spectrum. @@ -1180,7 +1369,7 @@ \subsection{Chained Three Body Parity Operators} \conssec \begin{itemize} -\item Does not reproduce the entire spectrum. +\item Reproduces energy spectrum, but not with the same degeneracy as the original Hamiltonian. \item Not very symmetric. \end{itemize} @@ -1230,6 +1419,80 @@ \subsection{Chained Three Body Parity Operators} \end{itemize} \newpage +%======= +% \subsection{Chained Three Body Parity Operators} +% +% \summarysec +% +% Goal: Guarantee that $z_{a_{k}}=z_iz_j$ can be chained together to make large product terms consisting of $z$. +% %Recall that products of $b$ correspond to parity check or ${\sc xor}$ operations, while products of $b$ correspond to logical ${\sc and}$ operations. +% The penalty term $P(z_{a_k},z_i,z_j)= \mp z_{a_k}z_iz_j$ guarantees that $z_{a_k}=\pm z_iz_j$. The 3-local term can be made from gadgets. +% +% This method was originally only used to reproduce the ground state of high locality terms, but states of the "wrong" parity ($q_k=\mp z_iz_j$) will all have the same energy as well, so it reproduces the full spectrum. +% +% +% \costsec +% \begin{itemize} +% \item The best known gadget for a 3-local Ising term uses one auxilliary qubit. Based on this gadget an $n\ge4$ body Ising term can be made using $3+2(n-4)$ auxilliary qubits. +% \end{itemize} +% +% \prossec +% \begin{itemize} +% \item Natural transmon implementation \cite{Leib2016}. +% \item Chain like structure means that long range connectivity not required. +% \end{itemize} +% +% \conssec +% \begin{itemize} +% \item Does not preserve degeneracy, ground state will retain orginal degeneracy, but excited states will have degeneracy multipled by $n-3$ +% \item Not very symmetric. +% \end{itemize} +% +% \vspace{-1mm} +% +% \examplesec +% \vspace{-4mm} +% +% \begin{align} +% \begin{gathered} +% H_\textrm{5-local} = z_1z_2z_3z_4z_5 =32\,b_1b_2b_3b_4b_5\nonumber \\ +% -16\,(b_1b_2b_3b_4+b_1b_2b_3b_5+b_1b_2b_4b_5+b_1b_3b_4b_5+b_2b_3b_4b_5) \nonumber \\ +% +8\,(b_1b_2b_3+b_1b_2b_4+b_1b_2b_5+b_1b_3b_4+b_1b_3b_5+b_1b_4b_5+b_2b_3b_4+b_2b_3b_4+b_2b_4b_5+b_3b_4b_5)\nonumber \\ +% -4\,(b_1b_2+b_1b_3+b_1b_4+b_1b_5+b_2b_3+b_2b_4+b_2b_5+b_3b_4+b_3b_5+b_4b_5)\nonumber \\ +% +2(b_1+b_2+b_3+b_4+b_5)-1 +% \end{gathered} +% \end{align} +% +% The full spectrum of $\pm z_1z_2z_3=\pm(8\,b_1b_2b_3-4\,b_1b_2-4\,b_2b_3+2\,b_1+2\,b_2+2\,b_3-1) $ is reproduced by: +% +% %\begin{equation} +% %P_\pm(z_1,z_2,z_3;\lambda)=\lambda\left(z_1z_2+z_2z_3+z_3z_1+2z_a(z_1+z_2+z_3)\right)\mp (z_1+z_2+z_3 +2z_a). +% %\end{equation} +% +% \begin{align} +% P_\pm(b_1,b_2,b_3;\lambda)=\lambda\left(4\,(b_1b_2+b_2b_3+b_3b_1)+8\,b_a(b_1+b_2+b_3)-12\,b_a-4\,(b_1+b_2+b_3)+3\right)\nonumber \\ +% \mp (2\,b_1+2\,b_2+2\,b_3 +4\,b_a-5). +% \end{align} +% +% This can be written more compactly in terms of $z$: +% +% \begin{equation} +% P_\pm(z_1,z_2,z_3;\lambda)=\lambda\left(z_1z_2+z_2z_3+z_3z_1+2z_a(z_1+z_2+z_3)\right)\mp (z_1+z_2+z_3 +2z_a). +% \end{equation} +% +% Using three copies of this 3-local gadget as a building block, and using two additional auxilliary variables, the spectrum of the 5-local term $z_1z_2z_3z_4z_5$ can be reproduced by the following Hamiltonian +% +% \begin{equation} +% H_{2-\rm{local}}=P_+(b_1,b_2,b_{a_1};\lambda)+P_+(b_{a_1},b_3,b_{a_2};\lambda)+P_+(b_{a_2},b_4,b_5;\lambda). +% \end{equation} +% +% \refsec +% \begin{itemize} +% \item Original proposal with transmon implementation: \cite{Leib2016}. +% \end{itemize} +% +% \newpage +%>>>>>>> 124543eb8af7ee94a551a5f88571879c749f8386 % \subsection{Multibody Operators in PAQC} % @@ -1306,6 +1569,7 @@ \subsection{Chained Three Body Parity Operators} \newpage + \part{\underline{{\normalsize Hamiltonians quadratic in $z$ and linear in $x$ (Transverse Field Ising Hamiltonians)}}\label{partTransverseIsing}} The Ising Hamiltonian with a transverse field in the $x$ direction is possible to implement in hardware: @@ -1463,107 +1727,292 @@ \part{\underline{{\normalsize{General Quantum Hamiltonians}}}}\label{partGeneral % %\end{comment} +\section{Non-perturbative Gadgets} -\section{$(3\rightarrow2)$ Gadgets} -The first gadgets for arbitrary Hamiltonians acting on some number of qubits, were designed to reproduce the spectrum of a 3-local Hamiltonian in the low-lying spectrum of a 2-local Hamiltonian. +\subsection{NP-OY (Ocko \& Yoshida, 2011)} -\subsection{$(3\rightarrow2)$-DC (Duan, Chen, 2011)} %The paper says "the terms in the product have to commute, but it's not clear if they mean each of the 3 terms of the product, or just the terms inside those terms, which would be much better. +\summarysec -% THERE IS A NEGATIVE SIGN DIFFERENCE FROM KKR, WHICH CAN'T BE ABSORVED INTO ANYTHING FOR THE SAME REASON WHY WE CAN'T ABSORB THE 6. IT'S BECAUSE THE 2-LOCAL HAMILTONIAN WILL HAVE TO ATTAIN THE NEGATIVE SIGN TOO, WHICH WE DON'T WANT BECAUSE WE JUST WANT TO WRITE THE 2-LOCAL TO 2-LOCAL TRANSFORMATION WITHOUT H_2LOCAL BEING PART OF THE FORUMA. NEGATING H_2LOCAL WOULD BE PART OF THE FORMULA OTHERWISE. +For the 8-body Hamiltonian: -\summarysec +%\begin{centering} +\begin{eqnarray} +\begin{gathered} +%\centering +%H_\textrm{8-local} = -J\sum_{ij} \left(x_{4i+3,j}x_{4i+2,j+1}x_{4i+4,j}x_{4i+5,j}x_{4i+7,j}x_{4i+6,j+1}x_{4i+4,j+1}x_{4i+5,j+1}\right.\\ +%\left.+ z_{4i+1,j}z_{4i+2,j}z_{4i+3,j}z_{4i+4,j} + z_{4i,j}z_{4i+1,j} + z_{4i+2,j}z_{4i+3,j-1} + z_{4i+3,j}z_{4i+3,j+1} + z_{4i+4,j}z_{4i+5,j}\right) +H_{8\textrm{-body}} =-J\sum_{ij}\left(x_{ij3}x_{ij+1,2}x_{ij4}x_{ij+1,4}x_{i+1j1}x_{i+1j+1,1}x_{i+1j3}x_{ij+1,2}+\right. \\ + \left.z_{ij1}z_{ij2}z_{ij3}z_{ij4}+z_{i-1j4}z_{ij1}+z_{ij2}z_{ij-1,3}+z_{ij4}z_{i+1j1}+z_{ij3}z_{ij+1,2}\right), +%\centering +\end{gathered} +\end{eqnarray} +%\end{centering} - For any group of 3-local terms that can be factored into a product of three 1-local factors, we can define three auxiliary qubits (regardless of the number of qubits we have in total) labeled by $a_{i}$ and make the transformation: -%Group all $3$-local terms together and express their sum as a sum of products of commuting matrices $s_{ij}$: +\noindent we define auxiliary qubits labeled by $a_{ijk}$, two auxiliaries for each pair $ij$: labeled $a_{ij1}$ and $a_{ij2}$. Then the 8-body Hamiltonian has the same low-lying eigenspace as the 4-body Hamiltonian: -%\begin{align} -%H_{3-\rm{local}} = \sum_i\prod_j^3 s_{ij} + H_{2-\rm{local}}. \label{eq:klocalInKempeMethod} -%\end{align} +%\begin{centering} +\begin{eqnarray} +\begin{gathered} +H_{4\textrm{-body}} =-\sum_{ij}\alpha\left(z_{ij1}z_{ij2}z_{ij3}z_{ij4}+z_{i,j,-1,4}z_{ij1}+z_{ij2}z_{i,j-1,3}+z_{ij4}z_{i+1,j,1}+z_{ij3}z_{i,j+1,2}\right.\\ + \left(1-z_{a_{ij1}}+z_{a_{ij2}}+z_{a_{ij1}}z_{a_{ij2}}\right)\left(z_{a_{i,j+1,1}}+z_{a_{i,j+1,2}}+z_{a_{i,j+1,1}}z_{a_{i,j+1,2}}-1\right)+\\ + \left.\left(1+z_{a_{ij1}}-z_{a_{ij2}}+z_{a_{ij1}}z_{a_{ij2}}\right)\left(1-z_{a_{i+1,j1}}-z_{a_{i+1,j2}}-z_{a_{i+1,j1}}z_{a_{i+1,j2}}\right)\right)+\\ + \frac{U}{2}\left(z_{a_{ij1}}+z_{a_{ij2}}+z_{a_{ij1}}z_{a_{ij2}}-1\right)+\\ + \frac{t}{2}\left(\left(x_{a_{ij2}}+z_{a_{ij1}}x_{a_{ij2}}\right)x_{ij3}x_{ij4}+\left(x_{a_{ij1}}x_{a_{ij2}}+y_{a_{ij1}}y_{a_{ij2}}\right)x_{i,j+1,2}x_{i,j+1,4}+\right.\\ +\left. \left.\left(x_{a_{ij2}}-z_{a_{ij1}}x_{a_{ij2}}\right)x_{i+1,j+1,1}x_{i+1,j+1,2}+\left(x_{a_{ij1}}x_{a_{ij2}}-y_{a_{ij1}}y_{a_{ij2}}\right)x_{i+1,j,1}x_{i+1,j,3}\right)\right). +\end{gathered} +\end{eqnarray} + +\noindent Now by defining the following ququits (spin-${3/2}$ particles, or 4-level systems): \begin{align} -\prod_i^3 \sum_j \alpha_{ij}s_{i} \rightarrow \alpha^I + \alpha_i^{ss} \sum_i \left(\sum_j \alpha_{ij}s_{ij}\right)^2 + \alpha_i^{sx}\sum_i \sum_j \alpha_{ij} s_{ij}x_{a_i} + \alpha^{zz} \sum_{ij} z_{a_i}z_{a_j} +s_{ijki^{\prime}j^{\prime}k^{\prime}}^{zz} &=z_{ijk}z_{i^{\prime}j^{\prime}k^{\prime}}\\ +s_{a_{ij}1}^{zz} &=\left(1-z_{a1_{ij}}+z_{a2_{ij}}+z_{a1_{ij}}z_{a2_{ij}}\right)\\ +s_{a_{ij}2}^{zz} &=\left(z_{a1_{ij}}+z_{a2_{ij}}+z_{a1_{ij}}z_{a2_{ij}}-1\right)\\ +s_{a_{ij}3}^{zz} &=\left(1+z_{a1_{ij}}-z_{a2_{ij}}+z_{a1_{ij}}z_{a2_{ij}}\right)\\ +s_{a_{ij}1}^{xz} &=\left(x_{a2_{ij}}+z_{a1_{ij}}x_{a2_{ij}}\right)\\ +s_{a_{ij}2}^{xz} &=\left(x_{a2_{ij}}-z_{a1_{ij}}x_{a2_{ij}}\right)\\ +s_{ijki^{\prime}j^{\prime}k^{\prime}}^{xx} &=x_{ijk}x_{i^{\prime}j^{\prime}k^{\prime}}\\ +s_{a_{ij}1}^{xy} &=\left(x_{a1_{ij}}x_{a2_{ij}}+y_{a1_{ij}}y_{a2_{ij}}\right)\\ +s_{a_{ij}2}^{xy} &=\left(x_{a1_{ij}}x_{a2_{ij}}-y_{a1_{ij}}y_{a2_{ij}}\right) \end{align} +\subsection*{NP-OY (Ocko \& Yoshida, 2011) [Continued]} +\noindent We can write the 4-body Hamiltonian on qubits as a 2-body Hamiltonian on ququits: -\begin{align} -\alpha^I &= \frac{1}{8\Delta} \\ -\alpha^{ss} &= \frac{1}{6\Delta^{\nicefrac{1}{3}}} \\ -\alpha^{sx} &= - \frac{1}{6\Delta^{\nicefrac{2}{3}}} \\ -\alpha^{zz} &= - \frac{1}{24\Delta} -\end{align} +{\scriptsize +\begin{eqnarray} +\begin{gathered} +H_{2\textrm{-body}} =-\sum_{ij}\left(\alpha\left(s_{ij1ij2}^{zz}s_{ij3ij4}^{zz}+s_{ij-1,4ij1}^{zz}+s_{ij2ij-1,3}^{zz}+s_{ij4i+1j1}^{zz}+s_{ij3ij+1,2}^{zz}+s_{a_{ij}1}^{zz}s_{a_{ij+1}2}^{zz}-s_{a_{ij}3}^{zz}s_{a_{i+1j}3}^{zz}\right)\right.\\ + \left.+\frac{U}{2}s_{a_{ij},1}^{zz}+\frac{t}{2}\left(s_{a_{ij}1}^{xz}s_{ij3ij4}^{xx}+s_{a_{ij}1}^{xy}s_{ij+1,2ij+1,4}^{xx}+s_{a_{ij}2}^{xz}s_{i+1,j+1,1i+1,j+1,2}^{xx}+s_{a_{ij}2}^{xy}s_{i+1j1,i+1,j3}^{xx}\right)\right). +\end{gathered} +\end{eqnarray} +} -\noindent The result will be a 2-local Hamiltonian whose low-lying spectrum is equivalent to the spectrum of $H_{3-\rm{local}}$ to within $\epsilon$ as long as $\Delta=\Theta\left(\epsilon^{-3}\right)$. +\noindent The low-lying eigenspace of $H_{2-\textrm{body}}$ is \textit{exactly} the same as for $H_{4-\textrm{local}}$. -\costsec +% \newpage +\costsec \begin{itemize} -\item 1 auxiliary qubit for each group of 3-local terms that can be factored into three 1-local factors. %\eqref{eq:klocalInKempeMethod}. -\item $\Delta =\Theta\left(\epsilon^{-3}\right)$ +\item 2 auxiliary ququits for each pair $ij$. +\item 6 more total terms (6 terms in the 8-body version becomes 12 terms: \\ +11 of them 2-body and 1 of them 1-body). \end{itemize} -\prossec + +\prossec \begin{itemize} -\item Very few auxiliary qubits needed +\item Non-perturbative. No prohibitive control precision requirement. +\item Only two auxiliaries required for each pair $ij$. +\item 8-body to 2-body transformation can be accmplished in 1 step, rather than a 1B1 gadget which would take 6 steps or an SD + $(3\rightarrow2)$ gadget combination which would take 4 steps. \end{itemize} - + \conssec \begin{itemize} -\item Will not work for Hamiltonians that do not factorize appropriately. -\end{itemize} - +\item Increase in dimention from working with only 2-level systems (spin-1/2 particles or $2\times2$ matrices) to working with 4-level systems (spin-3/2 particles). +\item Until now, only derived for a very specific Hamiltonian form. +\item This appraoch may become more demanding for Hamiltonians that are more than 8-local. +\end{itemize} + %\examplesec -% -%{\tiny -%\begin{align} -%x_1z_2y_3 - 3x_1x_2y_4 + z_1x_2 \rightarrow \alpha^I + \alpha^{zx}_{2a_{12}}z_2x_{a_12}+ \alpha^{xx}_{12} x_1x_2 + \alpha^{xx}_{1a_{11}}x_1x_{a_{11}} + \alpha^{xx}_{1a_{21}}x_1x_{a_{21}} + \alpha^{xx}_{2a_{22}}x_2x_{a_{22}} + \alpha^{yz}_{3a_{13}}y_3x_{a_{13}} + \alpha_{4a_{23}}^{yx}y_4x_{a_{23} } -%\end{align} -%} -% \refsec - \begin{itemize} -\item Original paper: \cite{Duan2011} +\item Original paper: \cite{Ocko2011}. \end{itemize} \newpage +\subsection{NP-SJ (Subasi \& Jarzynski, 2016)} +% CAN'T THIS BE DONE FO THE CASE WHERE THE \THETA IN EQ. 30 OF THEIR PAPER IS A SUM OF TERMS RATHER THAN JUST 1 TERM? -\subsection{$(3\rightarrow2)$-DC2 (Duan, Chen, 2011)} %The paper says s_i are Pauli matrices, but I'd bet that they just have to be commuting, well they don't have the s_i^2 terms, which would not be = 1 if they were general commuting matrices. +%Non-locality from the original Hamiltonian, is transfered into a unitary transformation operator, which when applied to a $k$-local Hamiltonian, can reduce the $k$-locality of high order terms at the expense of increasing the $k$-locality of other terms which were chosen to be at most $(k-2)-$local. \summarysec - For any 3-local term (product of Pauli matrices $s_i$) in the Hamiltonian, we can define \textit{one} auxiliary qubit labeled by $a$ and make the transformation: -%Group all $3$-local terms together and express their sum as a sum of products of commuting matrices $s_{ij}$: - -%\begin{align} -%H_{3-\rm{local}} = \sum_i\prod_j^3 s_{ij} + H_{2-\rm{local}}. \label{eq:klocalInKempeMethod} -%\end{align} +Determine the $k$-local term, $H_{k-\rm{local}}$, whose degree we wish to reduce, and factor it into two commuting factors: $H_{k^\prime-\rm{local}}H_{(k-k^\prime)-\rm{local}}$, where $k^\prime$ can be as low as 0. Separate all terms that are at momst $(k-1)$-local into ones that commmute with one of these factors (it does not matter which one, but without loss of generality we assume it to be the $(k-k^\prime)$-local one) and ones that anti-commute with it: \begin{align} -\alpha\prod_i^3 s_{i} \rightarrow \alpha^I + \alpha^s s_3 + \alpha^z z_a +\alpha^{ss} \left(s_1 + s_2 \right)^2 + \alpha^{sz}s_3z_a + + \alpha^{sx}\left(s_1x_a + s_2x_a \right) +%H_{k-\rm{local}} + H_{(k-1)\rm{-local}}^{\rm{commuting}} + H_{(k-1)\rm{-local}}^{\rm{anti-commuting}} + H_{ k)$-local! +\end{itemize} + + +\prossec +\begin{itemize} +\item Non-perturbative +\item Can linearize a term of arbitrary degree in one step. +\item Requires very few auxiliary qubits. +\end{itemize} + +\conssec +\begin{itemize} +\item Can introduce many new non-local terms as an expense for reducing only one $k$-local term. +\item If the portion of the Hamiltonian that does not commute with the $(k-k^\prime)$-local term has termms of degree $k-1$ (which can happen if $k^\prime=0$) they will all become $k$-local, so there is no guarantee that this method reduces $k$-locality. +\item If any terms were more than 1-local, this method will not fully quadratize the Hamiltonian (it must be combined with other methods). +\item It only works when the Hamiltonian's terms of degree at most $k-1$ all either commute or anti-commmute with the $k$-local term to be eliminated. +\end{itemize} + +\vspace{-1mm} + +\examplesec +\vspace{-3mm} + +\begin{align} +\vspace{-1mm} +4z_5 -3 x_1 + 2z_1y_2x_5 + 9x_1x_2x_3x_4 -x_1y_2z_3x_5 \rightarrow 9x_{a_{1}} + 4z_{a_4}z_5 -3z_{a_3}x_1 -z_{a_3}x_{a_4} a3 +2x_{a_3}x_5 +\end{align} + +\refsec +\begin{itemize} +\item Original paper, and description of the choices of terms and factors used for the given example \cite{Subas2016}. +\end{itemize} + +\subsection{NP-N (Nagaj, 2010)} + +%\summarysec + +%\costsec + +%\prossec +%\begin{itemize} +%\item +%\item +%\end{itemize} +% +%\conssec +%\begin{itemize} +%\item +%\item +%\end{itemize} + +%\examplesec + +\refsec +\begin{itemize} +\item Original paper: \cite{Nagaj2010}. +\end{itemize} + + + +\newpage +\section{Perturbative $(3\rightarrow2)$ Gadgets} + +The first gadgets for arbitrary Hamiltonians acting on some number of qubits, were designed to reproduce the spectrum of a 3-local Hamiltonian in the low-lying spectrum of a 2-local Hamiltonian. + +\subsection{P$(3\rightarrow2)$-DC (Duan, Chen, 2011)} %The paper says "the terms in the product have to commute, but it's not clear if they mean each of the 3 terms of the product, or just the terms inside those terms, which would be much better. + +% THERE IS A NEGATIVE SIGN DIFFERENCE FROM KKR, WHICH CAN'T BE ABSORVED INTO ANYTHING FOR THE SAME REASON WHY WE CAN'T ABSORB THE 6. IT'S BECAUSE THE 2-LOCAL HAMILTONIAN WILL HAVE TO ATTAIN THE NEGATIVE SIGN TOO, WHICH WE DON'T WANT BECAUSE WE JUST WANT TO WRITE THE 2-LOCAL TO 2-LOCAL TRANSFORMATION WITHOUT H_2LOCAL BEING PART OF THE FORUMA. NEGATING H_2LOCAL WOULD BE PART OF THE FORMULA OTHERWISE. + +\summarysec + + For any group of 3-local terms that can be factored into a product of three 1-local factors, we can define three auxiliary qubits (regardless of the number of qubits we have in total) labeled by $a_{i}$ and make the transformation: +%Group all $3$-local terms together and express their sum as a sum of products of commuting matrices $s_{ij}$: + +%\begin{align} +%H_{3-\rm{local}} = \sum_i\prod_j^3 s_{ij} + H_{2-\rm{local}}. \label{eq:klocalInKempeMethod} +%\end{align} + +\begin{align} +\prod_i^3 \sum_j \alpha_{ij}s_{i} \rightarrow \alpha^I + \alpha_i^{ss} \sum_i \left(\sum_j \alpha_{ij}s_{ij}\right)^2 + \alpha_i^{sx}\sum_i \sum_j \alpha_{ij} s_{ij}x_{a_i} + \alpha^{zz} \sum_{ij} z_{a_i}z_{a_j} +\end{align} + + +\begin{align} +\alpha^I &= \frac{1}{8\Delta} \\ +\alpha^{ss} &= \frac{1}{6\Delta^{\nicefrac{1}{3}}} \\ +\alpha^{sx} &= - \frac{1}{6\Delta^{\nicefrac{2}{3}}} \\ +\alpha^{zz} &= - \frac{1}{24\Delta} +\end{align} + +\noindent The result will be a 2-local Hamiltonian whose low-lying spectrum is equivalent to the spectrum of $H_{3-\rm{local}}$ to within $\epsilon$ as long as $\Delta=\Theta\left(\epsilon^{-3}\right)$. + +\costsec + +\begin{itemize} +\item 1 auxiliary qubit for each group of 3-local terms that can be factored into three 1-local factors. %\eqref{eq:klocalInKempeMethod}. +\item $\Delta =\Theta\left(\epsilon^{-3}\right)$ +\end{itemize} + +\prossec +\begin{itemize} +\item Very few auxiliary qubits needed +\end{itemize} + +\conssec +\begin{itemize} +\item Will not work for Hamiltonians that do not factorize appropriately. +\end{itemize} + +%\examplesec +% +%{\tiny +%\begin{align} +%x_1z_2y_3 - 3x_1x_2y_4 + z_1x_2 \rightarrow \alpha^I + \alpha^{zx}_{2a_{12}}z_2x_{a_12}+ \alpha^{xx}_{12} x_1x_2 + \alpha^{xx}_{1a_{11}}x_1x_{a_{11}} + \alpha^{xx}_{1a_{21}}x_1x_{a_{21}} + \alpha^{xx}_{2a_{22}}x_2x_{a_{22}} + \alpha^{yz}_{3a_{13}}y_3x_{a_{13}} + \alpha_{4a_{23}}^{yx}y_4x_{a_{23} } +%\end{align} +%} +% + +\refsec + +\begin{itemize} +\item Original paper: \cite{Duan2011} +\end{itemize} + + +\newpage + + + +\subsection{P$(3\rightarrow2)$-DC2 (Duan, Chen, 2011)} %The paper says s_i are Pauli matrices, but I'd bet that they just have to be commuting, well they don't have the s_i^2 terms, which would not be = 1 if they were general commuting matrices. + +\summarysec + + For any 3-local term (product of Pauli matrices $s_i$) in the Hamiltonian, we can define \textit{one} auxiliary qubit labeled by $a$ and make the transformation: +%Group all $3$-local terms together and express their sum as a sum of products of commuting matrices $s_{ij}$: + +%\begin{align} +%H_{3-\rm{local}} = \sum_i\prod_j^3 s_{ij} + H_{2-\rm{local}}. \label{eq:klocalInKempeMethod} +%\end{align} + +\begin{align} +\alpha\prod_i^3 s_{i} \rightarrow \alpha^I + \alpha^s s_3 + \alpha^z z_a +\alpha^{ss} \left(s_1 + s_2 \right)^2 + \alpha^{sz}s_3z_a + + \alpha^{sx}\left(s_1x_a + s_2x_a \right) +\end{align} + +{\scriptsize +\begin{align} +\alpha^I &= -\frac{1}{2\Delta} \\ +\alpha^s &= \alpha\left(\frac{1}{4\Delta^{\nicefrac{2}{3}}} - 1 \right) \\ +\alpha^z &= \alpha\left(\frac{1}{4\Delta^{\nicefrac{2}{3}}} - 1 \right) \\ +\alpha^{ss} &= \frac{1}{\Delta^{\nicefrac{1}{3}}} \\ +\alpha^{sz} &= \frac{\alpha}{4\Delta^{\nicefrac{2}{3}}} \\ +\alpha^{sx} &= \frac{1}{\Delta^{\nicefrac{2}{3}}} \\ +\end{align} +} + +\noindent The result will be a 2-local Hamiltonian whose low-lying spectrum is equivalent to the spectrum of $H_{3-\rm{local}}$ to within $\epsilon$ as long as $\Delta=\Theta\left(\epsilon^{-3}\right)$. + +\costsec + +\begin{itemize} +\item 1 auxiliary qubit for each 3-local term. %\eqref{eq:klocalInKempeMethod}. \item $\Delta =\Theta\left(\epsilon^{-3}\right)$ % need to check this. Assumed it. \end{itemize} @@ -1601,7 +2050,7 @@ \subsection{$(3\rightarrow2)$-DC2 (Duan, Chen, 2011)} %The paper says s_i are Pa \newpage -\subsection{$(3\rightarrow2)$-KKR (Kempe, Kitaev, Regev, 2004)} +\subsection{P$(3\rightarrow2)$-KKR (Kempe, Kitaev, Regev, 2004)} \summarysec @@ -1658,7 +2107,7 @@ \subsection{$(3\rightarrow2)$-KKR (Kempe, Kitaev, Regev, 2004)} \newpage -\subsection{$(3\rightarrow2)$-OT (Oliveira-Terhal, 2005)} +\subsection{P$(3\rightarrow2)$-OT (Oliveira-Terhal, 2005)} \summarysec @@ -1796,12 +2245,12 @@ \subsection{$(3\rightarrow2)$-OT (Oliveira-Terhal, 2005)} %% %\newpage -\section{1-by-1 Gadgets} +\section{Perturbative 1-by-1 Gadgets} A 1B1 gadget allows $k$-local terms to be quadratized one step at a time, where at each step the term's order is reduced by at most one. In each step, a $k$-local term is reduced to $\left(k-1\right)$-local, contrary to SD (sub-division) gadgets which can reduce $k$-local terms to $\left(\nicefrac{1}{2}\right)$-local in one step. -\subsection{1B1-OT (Oliveira \& Terhal, 2008)} +\subsection{P1B1-OT (Oliveira \& Terhal, 2008)} \summarysec @@ -1853,7 +2302,7 @@ \subsection{1B1-OT (Oliveira \& Terhal, 2008)} \newpage -\subsection{1B1-CBBK (Cao, Babbush, Biamonte, Kais, 2015)} +\subsection{P1B1-CBBK (Cao, Babbush, Biamonte, Kais, 2015)} \summarysec @@ -1895,13 +2344,13 @@ \subsection{1B1-CBBK (Cao, Babbush, Biamonte, Kais, 2015)} \end{itemize} \newpage -\section{Subdivision Gadgets} +\section{Perturbative Subdivision Gadgets} % !!!!!! Do we really need A and B to be non-overlapping, or is okay simply if they commute, which is a more general statement containing non-overlapping as a specific case? Instead of recursively reducing $k$-local to $(k-1)$-local one reduction at a time, we can reduce $k$-local terms to $(k/2)$-local terms directly for even $k$, or to $(k+1)/2$-local terms directly for odd $k$. Since when $k$ is odd we can add an identity operator to the $k$-local term to make it even, we will assume in the following that $k$ is even, in order to avoid having to write floor and ceiling functions. -\subsection{SD-OT (Oliveira \& Terhal, 2008)} +\subsection{PSD-OT (Oliveira \& Terhal, 2008)} \summarysec @@ -1990,7 +2439,7 @@ \subsection{SD-OT (Oliveira \& Terhal, 2008)} % %\newpage -\subsection{SD-CBBK (Cao, Babbush, Biamonte, Kais 2015)} +\subsection{PSD-CBBK (Cao, Babbush, Biamonte, Kais 2015)} \summarysec @@ -2135,7 +2584,6 @@ \subsection{SD-CBBK (Cao, Babbush, Biamonte, Kais 2015)} %\refsec \begin{comment} - \subsection{3-body Gadget from local X: Cao et al.~2013} Cao et al.~2013 produced two new gadgets and also generally improved the known constructions for several typical gadgets. @@ -2242,12 +2690,12 @@ \subsection{3-body Gadget from local X: Cao et al.~2013} \end{comment} \newpage -\section{Direct Reduction} +\section{Perturbative Direct Gadgets} Here we do not reduce $k$ by one order at a time (1B1 reduction) or by $\nicefrac{k}{2}$ at a time (SD reduction), but we directly reduce $k$-local terms to 2-local terms. % Instead of reducing $k$ in a one-by-one (1B1) way, or in a $\nicefrac{k}{2}$-by-$\nicefrac{k}{2}$ way as in sub-division (SD) gadgets, we reduce from $k$-local to 2-local in one step, using $k$ auxiliary qubits for each $k$-local term. -\subsection{DR-JF (Jordan \& Farhi, 2008)} +\subsection{PD-JF (Jordan \& Farhi, 2008)} \summarysec @@ -2289,7 +2737,7 @@ \subsection{DR-JF (Jordan \& Farhi, 2008)} \newpage -\subsection{DR-BFBD (Brell, Flammia, Bartlett, Doherty, 2011)} +\subsection{PD-BFBD (Brell, Flammia, Bartlett, Doherty, 2011)} \summarysec @@ -2341,143 +2789,191 @@ \subsection{DR-BFBD (Brell, Flammia, Bartlett, Doherty, 2011)} \end{itemize} \newpage -\section{Non-perturbative embeddings} -\subsection{NP-N (Nagaj, 2010)} -\summarysec -\costsec -%\prossec -%\begin{itemize} -%\item -%\item -%\end{itemize} -% -%\conssec -%\begin{itemize} -%\item -%\item -%\end{itemize} -%\examplesec -\refsec -\begin{itemize} -\item Original paper: \cite{Nagaj2010}. -\end{itemize} -\newpage -\subsection{NP-OY (Ocko \& Yoshida, 2011)} -\summarysec -The 8-body Hamiltonian: -%\begin{centering} -\begin{eqnarray} -\begin{gathered} -%\centering -H_\textrm{8-local} = -J\sum_{ij} \left(x_{4i+3,j}x_{4i+2,j+1}x_{4i+4,j}x_{4i+5,j}x_{4i+7,j}x_{4i+6,j+1}x_{4i+4,j+1}x_{4i+5,j+1}\right.\\ -\left.+ z_{4i+1,j}z_{4i+2,j}z_{4i+3,j}z_{4i+4,j} + z_{4i,j}z_{4i+1,j} + z_{4i+2,j}z_{4i+3,j-1} + z_{4i+3,j}z_{4i+3,j+1} + z_{4i+4,j}z_{4i+5,j}\right) -%\centering -\end{gathered} -\end{eqnarray} -%\end{centering} -\noindent is transformed into the following 2-body Hamiltonian: -\begin{eqnarray} -% -\end{eqnarray} -\costsec -%\prossec -%\begin{itemize} -%\item -%\item -%\end{itemize} -% -%\conssec -%\begin{itemize} -%\item -%\item -%\end{itemize} -%\examplesec -\refsec -\begin{itemize} -\item Original paper: \cite{Nagaj2010}. -\end{itemize} -\newpage -\subsection{NP-SJ (Subasi \& Jarzynski, 2016)} -% CAN'T THIS BE DONE FO THE CASE WHERE THE \THETA IN EQ. 30 OF THEIR PAPER IS A SUM OF TERMS RATHER THAN JUST 1 TERM? -%Non-locality from the original Hamiltonian, is transfered into a unitary transformation operator, which when applied to a $k$-local Hamiltonian, can reduce the $k$-locality of high order terms at the expense of increasing the $k$-locality of other terms which were chosen to be at most $(k-2)-$local. -\summarysec -Determine the $k$-local term, $H_{k-\rm{local}}$, whose degree we wish to reduce, and factor it into two commuting factors: $H_{k^\prime-\rm{local}}H_{(k-k^\prime)-\rm{local}}$, where $k^\prime$ can be as low as 0. Separate all terms that are at momst $(k-1)$-local into ones that commmute with one of these factors (it does not matter which one, but without loss of generality we assume it to be the $(k-k^\prime)$-local one) and ones that anti-commute with it: -\begin{align} -%H_{k-\rm{local}} + H_{(k-1)\rm{-local}}^{\rm{commuting}} + H_{(k-1)\rm{-local}}^{\rm{anti-commuting}} - H_{ k)$-local! -\end{itemize} -\prossec -\begin{itemize} -\item Non-perturbative -\item Can linearize a term of arbitrary degree in one step. -\item Requires very few auxiliary qubits. -\end{itemize} -\conssec -\begin{itemize} -\item Can introduce many new non-local terms as an expense for reducing only one $k$-local term. -\item If the portion of the Hamiltonian that does not commute with the $(k-k^\prime)$-local term has termms of degree $k-1$ (which can happen if $k^\prime=0$) they will all become $k$-local, so there is no guarantee that this method reduces $k$-locality. -\item If any terms were more than 1-local, this method will not fully quadratize the Hamiltonian (it must be combined with other methods). -\item It only works when the Hamiltonian's terms of degree at most $k-1$ all either commute or anti-commmute with the $k$-local term to be eliminated. -\end{itemize} -\vspace{-1mm} -\examplesec -\vspace{-3mm} -\begin{align} -\vspace{-1mm} -4z_5 -3 x_1 + 2z_1y_2x_5 + 9x_1x_2x_3x_4 -x_1y_2z_3x_5 \rightarrow 9x_{a_{1}} + 4z_{a_4}z_5 -3z_{a_3}x_1 -z_{a_3}x_{a_4} a3 +2x_{a_3}x_5 -\end{align} -\refsec -\begin{itemize} -\item Original paper, and description of the choices of terms and factors used for the given example \cite{Subas2016}. -\end{itemize} + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \newpage @@ -2575,6 +3071,7 @@ \section{Further References} \item More toric code gadgets: \cite{Brell2014}. \item Parity adiabatic quantum computing (LHZ lattice): \cite{Lechner2015}. \item Extensions of the LHZ scheme: \cite{Rocchetto2016}. +\item Minimizing $k$-local discrete functions with the help of continuous variable calculus\cite{Shen2017a}. \end{itemize} \newpage @@ -2597,6 +3094,11 @@ \section*{Acknowledgments} %Alex McCaskey, since in recent times I've been working hard on this to give him something easy to work with ??? % Courtney from the Toric codes. % Terhal +% Ocko +% Nagaj +% Charles, Zabih +% More Ishikawa +% More Gruber? % 2-> 2 gadgets: Oliveira-Terhal's paper describe the cross-gadget, fork-gadget, and triangle-gadget.