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

Abstract

We investigate the problem of synthesizing T-depth optimal quantum circuits for exactly implementable unitaries over the Clifford+T gate set. We construct a subset, $${{\mathbb{V}}}_{n}$$ V n , of T-depth 1 unitaries. T-depth-optimal decomposition of unitary U is $${e}^{i\phi }\left({\prod }_{i}{V}_{i}\right)C$$ e i ϕ ∏ i V i C , $${V}_{i}\in {{\mathbb{V}}}_{n}$$ V i ∈ V n , C is Clifford and $$| {{\mathbb{V}}}_{n}| \,\le \,n\cdot {2}^{5.6n}$$ ∣ V n ∣ ≤ n ⋅ 2 5.6 n . We use nested meet-in-the-middle technique to synthesize provably depth-optimal and T-depth-optimal circuits. For the latter, we achieve space and time complexity $$O({({4}^{{n}^{2}})}^{\lceil d/c\rceil })$$ O ( ( 4 n 2 ) ⌈ d / c ⌉ ) and $$O({({4}^{{n}^{2}})}^{(c-1)\lceil d/c\rceil })$$ O ( ( 4 n 2 ) ( c − 1 ) ⌈ d / c ⌉ ) respectively (d is the minimum T-depth, c ≥ 2 a constant). The previous best algorithm had complexity $$O({({3}^{n}\cdot {2}^{k{n}^{2}})}^{\lceil \frac{d}{2}\rceil }\cdot {2}^{k{n}^{2}})$$ O ( ( 3 n ⋅ 2 k n 2 ) ⌈ d 2 ⌉ ⋅ 2 k n 2 ) (k > 2.5 a constant). We design a more efficient algorithm with space and time complexity poly(n, 25.6n, d) (or $${{{\rm{poly}}}}({n}^{\log n},{2}^{5.6n},d)$$ poly ( n log n , 2 5.6 n , d ) with weaker assumptions). The claimed efficiency, optimality depends on conjectures.

T-count and T-depth of any multi-qubit unitary

“…Then we can perform amplitude test (Theorem 3.1) and conjugation test (Theorem 3.3, Corollary 3.1) and get an ϵ-A-count-optimal or ϵ-A-depth-optimal decomposition of V A . So it will be interesting and useful to find such nice generating set for other bases, as has been found for Clifford+T 25 (T-count), (T-depth) 27 , and Clifford+CS 34 (CS-count, only for 2-qubit unitaries). One simple way of constructing G for count-optimality is to write U A as follows.…”

“…It is not hard to see that if ϵ = 0 then we get the problem of synthesizing T-count and T-depth-optimal circuits for exactly implementable unitaries. In this case, both provable [25][26][27] and much more efficient heuristic 26,27 algorithms have been developed (see Table 1 for a comparison). We say that an algorithm is provable if its claimed efficiency and correctness or quality of solution (in this case optimality) can be proved by rigorous arguments.…”

“…For example, the T-depth-optimal synthesis algorithm of ref. 27 was able to generate optimal circuits for standard unitaries like Fredkin, Peres, and Quantum OR, which could not be done by the re-synthesis methods used in ref. 33 .…”

“…A generating set for depthoptimality can be constructed by conjugating products of at least n A-gates on distinct qubits by Clifford, as has been done in ref. 27 .…”

“…From Table 1 we see that for exactly implementable unitaries the provable algorithms like refs. [25][26][27] had an exponential dependence on T-count and T-depth. Significant improvements were achieved in refs.…”