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Quantum Compiling Methods for Fault-Tolerant Gate Sets of Dimension Greater than Two

Andrew N. Glaudell

Published Jan 1, 2019• 2 citations
Publisher

Abstract

Fault-tolerant gate sets whose generators belong to the Clifford hierarchy form the basis of many protocols for scalable quantum computing architectures. At the beginning of the decade, number-theoretic techniques were employed to analyze circuits over these gate sets on single qubits, providing the basis for a number of state-of-the-art quantum compiling algorithms. In this dissertation, I further this program by employing number-theoretic techniques for higher-dimensional gate sets on both qudit and multi-qubit circuits. First, I introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. I show that these canonical forms are T-optimal and describe an algorithm which takes as input a Clifford+T circuit and outputs the canonical form for that operator. The algorithm runs in time linear in the number of gates of the circuit. Our results provide a higher-dimensional generalization of prior work by Matsumoto and Amano who introduced similar canonical forms for single-qubit Clifford+T circuits. Finally, we show that a similar extension of these normal forms to higher dimensions exists, but do not establish uniqueness. Moving to multi-qubit circuits, I provide number-theoretic characterizations for certain restricted Clifford+T circuits by considering unitary matrices over subrings of Z[1/√2, i]. We focus on the subrings Z[1/2], Z[1/√2], Z[1/√−2], and Z[1/2, i], and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X, CX, CCX} with an analogue of the Hadamard gate and an optional phase gate. I then establish the existence and uniqueness of a normal form for one of these gate sets, the two-qubit gate set of Clifford+Controlled Phase gate CS. This normal form is optimal in the number of CS gates, making it the first normal form that is non-Clifford optimal for a fault tolerant universal multi-qubit gate set. We provide a synthesis algorithm that runs in a time linear in the gate count and outputs the equivalent normal form. In proving the existence and uniqueness of the normal form, we likewise establish the generators and relations for the two-qubit Clifford+CS group. Finally, we demonstrate that a lower bound of 5 log2 (1/ε) + O(1) CS gates are required to ε-approximate any 4 × 4 unitary matrix. Lastly, using the characterization of circuits over the Clifford+CS gate set and the existence of an optimal normal form, I provide an ancilla-free inexact synthesis algorithm for two-qubit unitaries using the Clifford+SC gate set for Pauli-rotations. These operators require 6 log2 (1/ε) + O(1) CS gates to synthesize in the typical case and 8 log2 (1/ε) + O(1) in the worst case.

DOI
10.13016/ehlg-4h1q

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Fault-tolerant gate sets whose generators belong to the Clifford hierarchy form the basis of many protocols for scalable quantum computing architectures. At the beginning of the decade, number-theoretic techniques were employed to analyze circuits over these gate sets on single qubits, providing the basis for a number of state-of-the-art quantum compiling algorithms. In this dissertation, I further this program by employing number-theoretic techniques for higher-dimensional gate sets on both qudit and multi-qubit circuits. First, I introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. I show that these canonical forms are T-optimal and describe an algorithm which takes as input a Clifford+T circuit and outputs the canonical form for that operator. The algorithm runs in time linear in the number of gates of the circuit. Our results provide a higher-dimensional generalization of prior work by Matsumoto and Amano who introduced similar canonical forms for single-qubit Clifford+T circuits. Finally, we show that a similar extension of these normal forms to higher dimensions exists, but do not establish uniqueness. Moving to multi-qubit circuits, I provide number-theoretic characterizations for certain restricted Clifford+T circuits by considering unitary matrices over subrings of Z[1/√2, i]. We focus on the subrings Z[1/2], Z[1/√2], Z[1/√−2], and Z[1/2, i], and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X, CX, CCX} with an analogue of the Hadamard gate and an optional phase gate. I then establish the existence and uniqueness of a normal form for one of these gate sets, the two-qubit gate set of Clifford+Controlled Phase gate CS. This normal form is optimal in the number of CS gates, making it the first normal form that is non-Clifford optimal for a fault tolerant universal multi-qubit gate set. We provide a synthesis algorithm that runs in a time linear in the gate count and outputs the equivalent normal form. In proving the existence and uniqueness of the normal form, we likewise establish the generators and relations for the two-qubit Clifford+CS group. Finally, we demonstrate that a lower bound of 5 log2 (1/ε) + O(1) CS gates are required to ε-approximate any 4 × 4 unitary matrix. Lastly, using the characterization of circuits over the Clifford+CS gate set and the existence of an optimal normal form, I provide an ancilla-free inexact synthesis algorithm for two-qubit unitaries using the Clifford+SC gate set for Pauli-rotations. These operators require 6 log2 (1/ε) + O(1) CS gates to synthesize in the typical case and 8 log2 (1/ε) + O(1) in the worst case.
Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2×2 unitary matrix V can be exactly represented by a single-qubit Clifford+ T circuit if and only if the entries of V belong to the ring Z[1/2,i] . Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+ T circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+ T circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+ T circuits by considering unitary matrices over subrings of Z[1/2,i] . We focus on the subrings Z[1/2] , Z[1/2] , Z[1/i2] , and Z[1/2,i] , and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X,CX,CCX} with an analogue of the Hadamard gate and an optional phase gate.
Abstract We study two-qubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlled-phase gate CS = diag(1, 1, 1, i). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes through magic state distillation. Since non-Clifford gates are typically more expensive to perform in a fault-tolerant manner, it is often desirable to construct circuits that use few CS gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for two-qubit Clifford+CS operators. Our algorithm inputs a Clifford+CS operator U and outputs a Clifford+CS circuit for U, which uses the least possible number of CS gates. Because the algorithm is deterministic, the circuit it associates to a Clifford+CS operator can be viewed as a normal form for that operator. We give an explicit description of these normal forms and use this description to derive a worst-case lower bound of 5log2(1ϵ)+O(1)5{{\rm{log}}}_{2}(\frac{1}{\epsilon })+O(1)5log2​(ϵ1​)+O(1) 5 log 2 ( 1 ϵ ) + O ( 1 ) on the number of CS gates required to ϵ-approximate elements of SU(4). Our work leverages a wide variety of mathematical tools that may find further applications in the study of fault-tolerant quantum circuits.
The efficiency of compiling high-level quantum algorithms into instruction sets native to quantum computers defines the moment in the future when we will be able to solve interesting and important problems on quantum computers. In my work I focus on the new methods for compiling single qubit operations that appear in many quantum algorithms into single qubit operations natively supported by several popular architectures. In addition, I study several questions related to synthesis and optimization of multiqubit operations. \n \n \nWhen studying the single qubit case, I consider two native instruction sets. The first one is Clifford+T; it is supported by conventional quantum computers implementing fault tolerance protocols based on concatenated and surface codes, and by topological quantum computers based on Ising anyons. The second instruction set is the one supported by topological quantum computers based on Fibonacci anyons. I show that in both cases one can use the number theoretic structure of the problem and methods of computational algebraic number theory to achieve improvements over the previous state of the art by factors ranging from 10 to 1000 for instances of the problem interesting in practice. This order of improvement might make certain interesting quantum computations possible several years earlier. \n \n \nThe work related to multiqubit operations is on exact synthesis and optimization of Clifford+T and Clifford circuits. I show an exact synthesis algorithm for unitaries generated by Clifford+T circuits requiring exponentially less number of gates than previous state of the art. For Clifford circuits two directions are studied: the algorithm for finding optimal circuits acting on a small number of qubits and heuristics for larger circuits optimization. The techniques developed allows one to reduce the size of encoding and decoding circuits for quantum error correcting codes by 40-50\\% and also finds their applications in randomized benchmarking protocols.
Abstract In quantum computing there are quite a few universal gate sets, each having their own characteristics. In this paper we study the Clifford+CS universal fault-tolerant gate set. The CS gate is used is many applications and this gate set is an important alternative to Clifford+T. We introduce a generating set in order to represent any unitary implementable by this gate set and with this we derive a bound on the CS-count of arbitrary multi-qubit unitaries. Analysing the channel representation of the generating set elements, we infer JnCS⊂JnT{\mathcal {J}}_n^{CS}\subset {\mathcal {J}}_n^TJnCS​⊂JnT​ J n CS ⊂ J n T , where JnCS{\mathcal {J}}_n^{CS}JnCS​ J n CS and JnT{\mathcal {J}}_n^TJnT​ J n T are the set of unitaries exactly implementable by the Clifford+CS and Clifford+T gate sets, respectively. We develop CS-count optimal synthesis algorithms for both approximately and exactly implementable multi-qubit unitaries. With the help of these we derive a CS-count-optimal circuit for Toffoli, implying JnTof=JnCS{\mathcal {J}}_n^{Tof}={\mathcal {J}}_n^{CS}JnTof​=JnCS​ J n Tof = J n CS , where JnTof{\mathcal {J}}_n^{Tof}JnTof​ J n Tof is the set of unitaries exactly implementable by the Clifford+Toffoli gate set. Such conclusions can have an important impact on resource estimates of quantum algorithms.
In this paper we study the universal V-basis gate sets, which have also been shown to be fault tolerant. Our methods and results can be applied to arbitrary dimensional basis gates, but we explicitly state results for basis gates for SU(2) and SU(4). We also include Cliffords, as done in earlier works. We introduce generating sets in order to represent any unitary implementable by these gate sets and with these we derive a bound on the V count of arbitrary multiqubit unitaries. We analyze the channel representation of the generating set elements, with the help of which we infer that none of these basis gates can be implemented exactly with the other universal fault-tolerant gate sets like Clifford+T, Clifford+CS, Clifford+Toffoli. Also T, CS, and Toffoli cannot be exactly implemented in the V basis. In fact, basis unitaries in dimension four cannot be implemented exactly with those in dimension two. We develop V-count-optimal synthesis algorithms for both approximately and exactly implementable multiqubit unitaries. With the help of these we show that two basis unitaries in dimension four are required to implement each basis unitary in dimension two. The space and time complexities of our provable algorithms are exponential in V count. But for the special case of one-qubit unitaries we achieve a complexity that is linear in V count. Both space and time complexities of our heuristic algorithms are polynomial in V count.
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.