A (quasi-)polynomial time heuristic algorithm for synthesizing T-depth optimal circuits

• URL: http://arxiv.org/abs/2101.03142v6
• Date: Tue, 13 Sep 2022 10:16:07 GMT
• Title: A (quasi-)polynomial time heuristic algorithm for synthesizing T-depth optimal circuits
• Authors: Vlad Gheorghiu, Michele Mosca, Priyanka Mukhopadhyay
• Abstract summary: We use nested meet-in-the-middle technique to develop algorithms for synthesizing provably emphdepth-optimal and emphT-depth-optimal circuits. For synthesizing T-depth-optimal circuits, we get an algorithm with space and time complexity $Oleft(left(4n2right)lceil d/crceilright)$. Our algorithm has space and time complexity $poly(n,25.6n,d)$ (or $poly(nlog n,
• Score: 1.933681537640272
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: We investigate the problem of synthesizing T-depth optimal quantum circuits over the Clifford+T gate set. First we construct a special subset of T-depth 1 unitaries, such that it is possible to express the T-depth-optimal decomposition of any unitary as product of unitaries from this subset and a Clifford (up to global phase). The cardinality of this subset is at most $n\cdot 2^{5.6n}$. We use nested meet-in-the-middle (MITM) technique to develop algorithms for synthesizing provably \emph{depth-optimal} and \emph{T-depth-optimal} circuits for exactly implementable unitaries. Specifically, for synthesizing T-depth-optimal circuits, we get an algorithm with space and time complexity $O\left(\left(4^{n^2}\right)^{\lceil d/c\rceil}\right)$ and $O\left(\left(4^{n^2}\right)^{(c-1)\lceil d/c\rceil}\right)$ respectively, where $d$ is the minimum T-depth and $c\geq 2$ is a constant. This is much better than the complexity of the algorithm by Amy et al.(2013), the previous best with a complexity $O\left(\left(3^n\cdot 2^{kn^2}\right)^{\lceil \frac{d}{2}\rceil}\cdot 2^{kn^2}\right)$, where $k>2.5$ is a constant. We design an even more efficient algorithm for synthesizing T-depth-optimal circuits. The claimed efficiency and optimality depends on some conjectures, which have been inspired from the work of Mosca and Mukhopadhyay (2020). To the best of our knowledge, the conjectures are not related to the previous work. Our algorithm has space and time complexity $poly(n,2^{5.6n},d)$ (or $poly(n^{\log n},2^{5.6n},d)$ under some weaker assumptions).

Related papers

• Nearly Optimal Regret for Decentralized Online Convex Optimization [53.433398074919]
  Decentralized online convex optimization (D-OCO) aims to minimize a sequence of global loss functions using only local computations and communications. We develop novel D-OCO algorithms that can respectively reduce the regret bounds for convex and strongly convex functions. Our algorithms are nearly optimal in terms of $T$, $n$, and $rho$.
  arXiv  Detail & Related papers  (2024-02-14T13:44:16Z)
• Dueling Optimization with a Monotone Adversary [35.850072415395395]
  We study the problem of dueling optimization with a monotone adversary, which is a generalization of (noiseless) dueling convex optimization. The goal is to design an online algorithm to find a minimizer $mathbfx*$ for a function $fcolon X to mathbbRd.
  arXiv  Detail & Related papers  (2023-11-18T23:55:59Z)
• An Oblivious Stochastic Composite Optimization Algorithm for Eigenvalue Optimization Problems [76.2042837251496]
  We introduce two oblivious mirror descent algorithms based on a complementary composite setting. Remarkably, both algorithms work without prior knowledge of the Lipschitz constant or smoothness of the objective function. We show how to extend our framework to scale and demonstrate the efficiency and robustness of our methods on large scale semidefinite programs.
  arXiv  Detail & Related papers  (2023-06-30T08:34:29Z)
• ReSQueing Parallel and Private Stochastic Convex Optimization [59.53297063174519]
  We introduce a new tool for BFG convex optimization (SCO): a Reweighted Query (ReSQue) estimator for the gradient of a function convolved with a (Gaussian) probability density. We develop algorithms achieving state-of-the-art complexities for SCO in parallel and private settings.
  arXiv  Detail & Related papers  (2023-01-01T18:51:29Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• T-count and T-depth of any multi-qubit unitary [1.933681537640272]
  We design provable algorithm to determine T-count of any $n$-qubit ($ngeq 1$) unitary $W$ of size $2ntimes 2n$, over the Clifford+T gate set. Our algorithm can also be used to determine the (minimum possible) T-depth of any multi-qubit unitary.
  arXiv  Detail & Related papers  (2021-10-19T22:16:00Z)
• Hybrid Stochastic-Deterministic Minibatch Proximal Gradient: Less-Than-Single-Pass Optimization with Nearly Optimal Generalization [83.80460802169999]
  We show that HSDMPG can attain an $mathcalObig (1/sttnbig)$ which is at the order of excess error on a learning model. For loss factors, we prove that HSDMPG can attain an $mathcalObig (1/sttnbig)$ which is at the order of excess error on a learning model.
  arXiv  Detail & Related papers  (2020-09-18T02:18:44Z)
• Streaming Complexity of SVMs [110.63976030971106]
  We study the space complexity of solving the bias-regularized SVM problem in the streaming model. We show that for both problems, for dimensions of $frac1lambdaepsilon$, one can obtain streaming algorithms with spacely smaller than $frac1lambdaepsilon$.
  arXiv  Detail & Related papers  (2020-07-07T17:10:00Z)
• A polynomial time and space heuristic algorithm for T-count [2.28438857884398]
  This work focuses on reducing the physical cost of implementing quantum algorithms when using the state-of-the-art fault-tolerant quantum error correcting codes. We consider the group of unitaries that can be exactly implemented by a quantum circuit consisting of the Clifford+T gate set, a universal gate set.
  arXiv  Detail & Related papers  (2020-06-22T17:21:41Z)
• Improved quantum algorithm for A-optimal projection [4.248054546466641]
  This paper corrects the time complexity of Duan emphet al.'s algorithm to $(frackappa4ssqrtks epsilonsmathrmpolylog)$. Our algorithm achieves at least a speedup compared to Duan emphet al.'s algorithm.
  arXiv  Detail & Related papers  (2020-06-10T09:31:53Z)
• Statistically Efficient, Polynomial Time Algorithms for Combinatorial Semi Bandits [3.9919781077062497]
  We consider semi-bandits over a set of arms $cal X subset 0,1d$ where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields the smallest known regret bound $R(T) = cal OBig( d (ln m)2 (ln T) over Delta_min Big)$, but it has computational complexity $cal O(|cal X|)$ which is typically exponential in $d$, and cannot
  arXiv  Detail & Related papers  (2020-02-17T21:32:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.