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

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 $c\ge 2$ is a constant. This is much better than the complexity of the algorithm by Amy et al.(2013), the previous best with a complexity $O\left({\left(3^n\cdot 2^{k{n}^2}\right)}^{\lceil \frac{d}{2}\rceil }\cdot 2^{k{n}^2}\right)$, 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).

arXiv commentPublished in Nature Partner Journal Quantum Information. Compared to v4: Minor changes. Not the exact journal version

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• T-count and T-depth of any multi-qubit unitary
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• Synthesizing efficient circuits for Hamiltonian simulation
• Assessment of various Hamiltonian partitionings for the electronic structure problem on a quantum computer using the Trotter approximation
• Lower T-count with faster algorithms
• A quantum random access memory (QRAM) using a polynomial encoding of binary strings
• Composability of global phase invariant distance and its application to approximation error management