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arXiv:2012.09224

Reducing stabilizer circuits without the symplectic group

Marc Bataille

Published Dec 16, 2020• 0 citations
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Abstract

We start by studying the subgroup structures underlying stabilizer circuits. Then we apply our results to provide two normal forms for stabilizer circuits. These forms are computed by induction using simple conjugation rules in the Clifford group and our algorithms do not rely on a special decomposition in the symplectic group. The first normal form has shape CX-CZ-P-Z-X-H-CZ-P-H, where CX (resp. CZ) denotes a layer of CNOT (resp. controlled-Z) gates, P a layer of phase gates, X (resp. Z) a layer of Pauli-X (resp. Pauli-Z) gates. Then we replace most of the controlled-Z gates by CNOT gates to obtain a second normal form of type P-CX-CZ-CX-Z-X-H-CZ-CX-P-H. In this second form, both controlled-Z layers have depth 1 and together contain therefore at most n controlled-Z gates. We also consider normal forms for stabilizer states and graph states. Finally we carry out a few tests on classical and quantum computers in order to show experimentally the utility of these normal forms to reduce the gate count of a stabilizer circuit.

DOI
10.48550/arxiv.2012.09224

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We start by studying the subgroup structures underlying stabilizer circuits. Then we apply our results to provide two normal forms for stabilizer circuits. These forms are computed by induction using simple conjugation rules in the Clifford group and our algorithms do not rely on a special decomposition in the symplectic group. The first normal form has shape CX-CZ-P-Z-X-H-CZ-P-H, where CX (resp. CZ) denotes a layer of CNOT (resp. controlled-Z) gates, P a layer of phase gates, X (resp. Z) a layer of Pauli-X (resp. Pauli-Z) gates. Then we replace most of the controlled-Z gates by CNOT gates to obtain a second normal form of type P-CX-CZ-CX-Z-X-H-CZ-CX-P-H. In this second form, both controlled-Z layers have depth 1 and together contain therefore at most n controlled-Z gates. We also consider normal forms for stabilizer states and graph states. Finally we carry out a few tests on classical and quantum computers in order to show experimentally the utility of these normal forms to reduce the gate count of a stabilizer circuit.
We start by studying the subgroup structures underlying stabilizer circuits and we use our results to propose a new normal form for stabilizer circuits. This normal form is computed by induction using simple conjugation rules in the Clifford group. It has shape CX-CZ-P-H-CZ-P-H, where CX (resp. CZ) denotes a layer of $\cnot$ (resp. $\cz$) gates, P a layer of phase gates and H a layer of Hadamard gates. Then we consider a normal form for stabilizer states and we show how to reduce the two-qubit gate count in circuits implementing graph states. Finally we carry out a few numerical tests on classical and quantum computers in order to show the practical utility of our methods. All the algorithms described in the paper are implemented in the C language as a Linux command available on GitHub.
In this paper, we improve the layered implementation of arbitrary stabilizer circuits introduced by Aaronson and Gottesman in Phys. Rev. A 70 (052328), 2004: to implement a general stabilizer circuit, we reduce their 11-stage computation -H-C-P-C-P-C-H-P-C-P-Cover the gate set consisting of Hadamard, controlled-NOT, and phase gates, into a 7-stage computation of the form -C-CZ-P-H-P-CZ-C-. We show arguments in support of using -CZ- stages over the -C- stages: not only the use of -CZ- stages allows a shorter layered expression, but -CZ- stages are simpler and appear to be easier to implement compared to the -C- stages. Based on this decomposition, we develop a two-qubit gate depth(14n-4) implementation of stabilizer circuits over the gate library {H, P, CNOT}, executable in the Linear Nearest Neighbor (LNN) architecture, improving best previously known depth25n circuit, also executable in the LNN architecture. Our constructions rely on Bruhat decomposition of the symplectic group and on folding arbitrarily long sequences of the form (-P-C-)minto a three-stage computation -P-CZ-C-. Our results include the reduction of the 11-stage decomposition -H-C-P-C-PC-H-P-C-P-Cinto a 9-stage decomposition of the form -C-P-C-PH-C-P-C-P-. This reduction is based on the Bruhat decomposition of the symplectic group. This result also implies a new normal form for stabilizer circuits. We show that a circuit in this normal form is optimal in the number of Hadamard gates used. We also show that the normal form has an asymptotically optimal number of parameters.
In this paper we improve the layered implementation of arbitrary stabilizer circuits introduced by Aaronson and Gottesman in {\it Phys. Rev. A 70(052328)}, 2004. In particular, we reduce their 11-stage computation -H-C-P-C-P-C-H-P-C-P-C- into an 8-stage computation of the form -H-C-CZ-P-H-P-CZ-C-. We show arguments in support of using -CZ- stages over the -C- stages: not only the use of -CZ- stages allows a shorter layered expression, but -CZ- stages are simpler and appear to be easier to implement compared to the -C- stages. Relying on the 8-stage decomposition we develop a two-qubit depth-(14n−4)(14n-4)(14n−4) implementation of stabilizer circuits over the gate library {P,H,CNOT}, executable in the LNN architecture, improving best previously known depth-25n25n25n circuit, also executable in the LNN architecture. Our constructions rely on folding arbitrarily long sequences (((-P-C-)m)^m)m into a 3-stage computation -P-CZ-C-, as well as efficient implementation of the -CZ- stage circuits.
(Abridged abstract.) In this thesis we introduce new models of quantum computation to study the emergence of quantum speed-up in quantum computer algorithms. Our first contribution is a formalism of restricted quantum operations, named normalizer circuit formalism, based on algebraic extensions of the qubit Clifford gates (CNOT, Hadamard and π/4π/4π/4-phase gates): a normalizer circuit consists of quantum Fourier transforms (QFTs), automorphism gates and quadratic phase gates associated to a set GGG, which is either an abelian group or abelian hypergroup. Though Clifford circuits are efficiently classically simulable, we show that normalizer circuit models encompass Shor's celebrated factoring algorithm and the quantum algorithms for abelian Hidden Subgroup Problems. We develop classical-simulation techniques to characterize under which scenarios normalizer circuits provide quantum speed-ups. Finally, we devise new quantum algorithms for finding hidden hyperstructures. The results offer new insights into the source of quantum speed-ups for several algebraic problems. Our second contribution is an algebraic (group- and hypergroup-theoretic) framework for describing quantum many-body states and classically simulating quantum circuits. Our framework extends Gottesman's Pauli Stabilizer Formalism (PSF), wherein quantum states are written as joint eigenspaces of stabilizer groups of commuting Pauli operators: while the PSF is valid for qubit/qudit systems, our formalism can be applied to discrete- and continuous-variable systems, hybrid settings, and anyonic systems. These results enlarge the known families of quantum processes that can be efficiently classically simulated. This thesis also establishes a precise connection between Shor's quantum algorithm and the stabilizer formalism, revealing a common mathematical structure in several quantum speed-ups and error-correcting codes.
Improving the simulation of quantum circuits on classical computers is important for understanding quantum advantage and increasing development speed. In this paper, we explore a way to express stabilizer states and further improve the speed of simulating stabilizer circuits with a current existing approach. First, we discover a unique and elegant canonical form for stabilizer states based on graph states to better represent stabilizer states and show how to efficiently simplify stabilizer states to canonical form. Second, we develop an improved algorithm for graph state stabilizer simulation and establish limitations on reducing the quadratic runtime of applying controlled Pauli ZZZ gates. We do so by creating a simpler formula for combining two Pauli-related stabilizer states into one. Third, to better understand the linear dependence of stabilizer states, we characterize all linearly dependent triplets, revealing symmetries in the inner products. Using our controlled Pauli ZZZ algorithm, we improve runtime for inner product computation from O(n3)O({n}^{3})O(n3) to O(nd2)O(n{d}^{2})O(nd2), where ddd is the maximum degree of the graph encountered during the calculation.