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Iranian Conference on Electrical Engineering

An Efficient and Exact Synthesis of N-Qubit Diagonal Ring Gates into Clifford and T Gates

Soudabe Mohamadzade, Mehdi Sedighi, Morteza Saheb Zamani

Published May 1, 2018• 0 citations
Publisher

Abstract

Exact synthesis of arbitrary unitary quantum gates into a Fault Tolerant (FT) gate library is generally impossible due to the limitation of available libraries. However, exact synthesis methods for special quantum gates can be found in the literature. In this paper, two approaches for exact synthesis of n-qubit ring multiplexed- $R_{\text{z}}$ and n-qubit ring diagonal gates $\mathbb{Z}(\pmb{1}/\sqrt{2},\ i)$ are presented. The proposed approaches use QSD to synthesize diagonal and multiplexed- $R_{z}$ gates into intermediate gates and then synthesis those to generate single-qubit ring gates. By using QSD, an n-qubit ring diagonal gate is decomposed into CNOT and $R_{z}$ gates that have an exact synthesis into CTL. The results show that the proposed approach for the diagonal ring gates generates $O(2^{n+1})$ gates instead of $O(n9^{n})$ gates produced by other methods.

DOI
10.1109/icee.2018.8472618

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In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring Z[12,i]\mathbb{Z}[\frac{1}{\sqrt{2}},i]Z[2​1​,i], in the single-qubit case. We report an efficient synthesis algorithm, with an exact optimality guarantee on the number of Hadamard and T gates used. We conjecture that the equivalence of the sets of unitaries implementable by circuits over the Clifford and T library and unitaries over the ring Z[12,i]\mathbb{Z}[\frac{1}{\sqrt{2}},i]Z[2​1​,i] holds in the nnn-qubit case.
We describe a new method for approximating an arbitrary nnn qubit unitary with precision ε\varepsilonε using a Clifford and T circuit with O(4nn(log⁡(1/ε)+n))O(4^{n}n(\log(1/\varepsilon)+n))O(4nn(log(1/ε)+n)) gates. The method is based on rounding off a unitary to a unitary over the ring Z[i,1/2]\mathbb{Z}[i,1/\sqrt{2}]Z[i,1/2​] and employing exact synthesis. We also show that any nnn qubit unitary over the ring Z[i,1/2]\mathbb{Z}[i,1/\sqrt{2}]Z[i,1/2​] with entries of the form (a+b2+ic+id2)/2k(a+b\sqrt{2}+ic+id\sqrt{2})/2^{k}(a+b2​+ic+id2​)/2k can be exactly synthesized using O(4nnk)O(4^{n}nk)O(4nnk) Clifford and T gates using two ancillary qubits. This new exact synthesis algorithm is an improvement over the best known exact synthesis method by B. Giles and P. Selinger requiring O(32nnk)O(3^{2^{n}}nk)O(32nnk) elementary gates.
Matchgate unitaries are ubiquitous in quantum computation due to their relation to non-interacting fermions and because they can be used to benchmark quantum computers. Implementing such unitaries on fault-tolerant devices requires first compiling them into a discrete universal gate set, typically Clifford+T+T+T. Here, we propose a different approach for their synthesis: compile matchgate unitaries using only matchgate gates. To this end, we first show that the matchgate-Clifford group (the intersection of the matchgate and Clifford groups) plus the T‾\overline{T}T gate (a TTT unitary up to a phase) is universal for the matchgate group. Our approach leverages the connection between nnn-qubit matchgate circuits and the standard representation of SO(2n)\mathbb{SO}(2n)SO(2n), which reduces the compilation from 2n×2n2^n\times 2^n2n×2n unitaries to 2n×2n2n\times2n2n×2n ones, thus reducing exponentially the size of the target matrix. Moreover, we rigorously show that this scheme is efficient, as an approximation error εSO(2n)\varepsilon_{\mathbb{SO}(2n)}εSO(2n)​ incurred in this smaller-dimensional representation translates at most into an O(n εSO(2n))O(n \,\varepsilon_{\mathbb{SO}(2n)})O(nεSO(2n)​) error in the exponentially large unitary. In addition, we study the exact version of the matchgate synthesis problem, and we prove that all matchgate unitaries UUU such that U⊗U∗U\otimes U^*U⊗U∗ has entries in the ring Z[1/2,i]\mathbb{Z}\big[1/\sqrt 2,i\big]Z[1/2​,i] can be exactly synthesized by a finite sequence of gates from the matchgate-Clifford+T‾+\overline{T}+T set, without ancillas. We then use this insight to map optimal exact matchgate synthesis to Boolean satisfiability, and compile the circuits that diagonalize the free-fermionic XXXXXX Hamiltonian on n=4, 8n=4,\,8n=4,8 qubits.
In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring Z[12,i]\mathbb{Z}[\frac{1}{\sqrt{2}},i]Z[2​1​,i], in the single-qubit case. We report an efficient synthesis algorithm, with an exact optimality guarantee on the number of Hadamard gates used. We conjecture that the equivalence of the sets of unitaries implementable by circuits over the Clifford and T library and unitaries over the ring Z[12,i]\mathbb{Z}[\frac{1}{\sqrt{2}},i]Z[2​1​,i] holds in the nnn-qubit case.
Resource-efficient and high-precision approximate synthesis of quantum circuits expressed in the Clifford+T gate set is vital for Fault-Tolerant quantum computing.Efficient optimal methods are known for single-qubit R Z unitaries, otherwise the problem is generally intractable.Search-based methods, like simulated annealing, empirically generate low resource cost approximate implementations of general multi-qubit unitaries so long as low precision (Hilbert-Schmidt distances of ε ≥ 10 -2 ) can be tolerated.These algorithms build up circuits that directly invert target unitaries.We instead leverage search-based methods to first approximately diagonalize a unitary, then perform the inversion analytically.This lets difficult continuous rotations be bypassed and handled in a postprocessing step.Our approach improves both the implementation precision and run time of synthesis algorithms by orders of magnitude when evaluated on unitaries from real quantum algorithms.On benchmarks previously synthesizable only with analytical techniques like the Quantum Shannon Decomposition, diagonalization uses an average of 95% fewer non-Clifford gates.