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

Gheorghiu, Vlad; Mosca, Michele; Mukhopadhyay, Priyanka

doi:10.48550/arxiv.2101.03142

Cited by 6 publications

(16 citation statements)

References 15 publications

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“…Then we can perform amplitude test (Theorem 3.1) and conjugation test (Theorem 3.3, Corollary 3.1) and get an -A-count-optimal or -A-depth-optimal decomposition of V A . So for other bases the main task remains to find such nice generating set, as has been found for Clifford+T [GKMR14] (T-count), [GMM21] (T-depth) and Clifford+CS [GRT21] (CS-count, only for 2-qubit unitaries). One simple way of constructing G for count-optimality is to write U A as follows.…”

Section: Our Contributionsmentioning

confidence: 99%

“…But exploiting some other algebraic properties, maybe we can do much better. A generating set for depth-optimality can be constructed by conjugating products of at least n A-gates on distinct qubits by Clifford, as has been done in [GMM21]. We implemented our algorithm to obtain the T-count of controlled R z (θ) (cR z (θ)) gate, Givens rotation (G(θ)), 2 and 3-qubit Quantum Fourier Transform (QFT).…”

Section: Our Contributionsmentioning

confidence: 99%

“…From Table 1 we see that for exactly implementable unitaries the provable algorithms like [GKMR14, MM21, GMM21] had an exponential dependence on T-count and T-depth. Significant improvements were achieved in [MM21,GMM21], where heuristics were designed that led to algorithms with a polynomial dependence on T-count and T-depth. The algorithm in our paper also suffer from exponential dependence on -T-count and -T-depth, which usualy have an inverse dependence on i.e.…”

Section: Our Contributionsmentioning

confidence: 99%

“…However, considerable amount of work has been done to synthesize T-count and T-depthoptimal circuits for exactly implementable ( = 0) multi-qubit unitaries. These include algorithms with provable optimality like [GKMR14, MM21, GMM21] that employ meet-in-the-middle (MITM) and nested MITM techniques, as well as more efficient heuristic algorithms whose optimality depend on some conjecture [MM21,GMM21].…”

Section: Related Workmentioning

confidence: 99%

“…It is not hard to see that if = 0 then we get the problem of synthesizing T-count and T-depth-optimal circuits for exactly implementable unitaries. In this case, both provable [GKMR14, MM21, GMM21] and much more efficient heuristic [MM21,GMM21] algorithms have been developed (see Table 1 for a comparison).…”

Section: Introductionmentioning

confidence: 99%

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

Gheorghiu¹,

Mosca²,

Mukhopadhyay³

2021

Preprint

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While implementing a quantum algorithm it is crucial to reduce the quantum resources, in order to obtain the desired computational advantage. For most fault-tolerant quantum errorcorrecting codes the cost of implementing the non-Clifford gate is the highest among all the gates in a universal fault-tolerant gate set.In this paper we design provable algorithm to determine T-count of any n-qubit (n ≥ 1) unitary W of size 2 n × 2 n , over the Clifford+T gate set. The space and time complexity of our algorithm are poly(2 n ) and poly ((2 n ) m ) respectively. m ( -T-count) is the T-count of an exactly implementable unitary U i.e. T (U ), such that d(U, W ) ≤ and T (U ) ≤ T (U ) where U is any exactly implementable unitary with d(U , W ) ≤ . d(., .) is the global phase invariant distance. Our algorithm can also be used to determine T-depth of any multi-qubit unitary and the asymptotic complexity has similar dependence on n and -T-depth. This is the first algorithm that gives T-count or T-depth of any multi-qubit (n ≥ 1) unitary (considering any distance). For small enough , we can synthesize the T-count and T-depth-optimal circuits.Our results can be used to determine optimal count (or depth) of non-Clifford gates required to implement any multi-qubit unitary with a universal gate set consisting of Clifford and non-Clifford gates like Clifford+CS, Clifford+V, etc. To the best of our knowledge, there were no resource-optimal synthesis algorithm for arbitrary multi-qubit unitaries in any universal gate set.

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Section: Our Contributionsmentioning

confidence: 99%

Section: Our Contributionsmentioning

confidence: 99%

Section: Our Contributionsmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Gheorghiu¹,

Mosca²,

Mukhopadhyay³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

A polynomial time and space heuristic algorithm for T-count

Mosca¹,

Mukhopadhyay²

2021

Quantum Sci. Technol.

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An important part of reaping computational advantage from a quantum computer is to reduce the quantum resources needed to implement a desired quantum algorithm. Quantum algorithms that are too large to be practical on noisy intermediate scale quantum devices will require fault-tolerant error correction. 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, in particular, those for which implementing the T gate consumes vastly more resources than the other gates in the gate set. More specifically, in this paper we consider the group of unitaries that can be exactly implemented by a quantum circuit consisting of the Clifford + T gate set. The Clifford + T gate set is a universal gate set and in this group, using state-of-the-art surface codes, the T gate is by far the most expensive component to implement fault-tolerantly. So it is important to minimize the number of T gates necessary for a fault-tolerant implementation. Our primary interest is to compute a circuit for a given n-qubit unitary U, using the minimum possible number of T gates (called the T-count of unitary U). We consider the problem COUNT-T, the optimization version of which aims to find the T-count of U. In its decision version the goal is to decide if the T-count is at most some positive integer m. Given an oracle for COUNT-T, we can compute a T-count-optimal circuit in time polynomial in the T-count and dimension of U. We give a provable classical algorithm that solves COUNT-T (decision) in time O N 2(c−1) m c poly(m, N) and space O N 2 m c poly(m, N) , where N = 2 n and c 2. This gives a space-time trade-off for solving this problem with variants of meet-in-the-middle techniques. We also introduce an asymptotically faster multiplication method that shaves a factor of N 0.7457 off of the overall complexity. Lastly, beyond our improvements to the rigorous algorithm, we give a heuristic algorithm that outputs a T-count-optimal circuit and has space and time complexity poly(m, N), under some assumptions. In our heuristic algorithm we developed a novel way of pruning the search space. While our heuristic method still scales exponentially with the number of qubits (though with a lower exponent), there is a large improvement by going from exponential to polynomial scaling with m. We implemented our heuristic algorithm with up to 4 qubit unitaries and obtained a significant improvement in time. For all benchmark and random unitaries we studied, the T-count returned by our algorithm is at most the T-count of their circuits shown in previous papers. iπ 4 P.Roughly speaking, each R(P) can be implemented with a circuit consisting of only one T gate. More detail has been given in section 2.2. Operator P is an n-qubit non-identity Pauli operator (defined in section A.1). We developed a fast algorithm in section 3 that computes W in time O(N 4 ). Currently the fastest algorithm

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Phase polynomials synthesis algorithms for NISQ architectures and beyond

Vivien

Martiel

Brugière

2022

Quantum Sci. Technol.

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We present a framework for the synthesis of phase polynomials that addresses both cases of full connectivity and partial connectivity for NISQ architectures. In most cases, our algorithms generate circuits with lower CNOT count and CNOT depth than the state of the art or have a signiﬁcantly smaller running time for similar performances. We also provide methods that can be applied to our algorithms in order to trade an increase in the CNOT count for a decrease in execution time, thereby ﬁlling the gap between our algorithms and faster ones.

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

Cited by 6 publications

References 15 publications

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

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

A polynomial time and space heuristic algorithm for T-count

Phase polynomials synthesis algorithms for NISQ architectures and beyond

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