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arXiv:2403.06777 · Electronic Proceedings in Theoretical Computer Science

Fast Classical Simulation of Quantum Circuits via Parametric Rewriting in the ZX-Calculus

Matthew Sutcliffe, Aleks Kissinger

Published Aug 19, 2025• 1 citation
arXivHTMLPDF

Abstract

The ZX-calculus is an algebraic formalism that allows quantum computations to be simplified via a small number of simple graphical rewrite rules.Recently, it was shown that, when combined with a family of "sum-over-Cliffords" techniques, the ZX-calculus provides a powerful tool for classical simulation of quantum circuits.However, for several important classical simulation tasks, such as computing the probabilities associated with many measurement outcomes of a single quantum circuit, this technique results in reductions over many very similar diagrams, where much of the same computational work is repeated.In this paper, we show that the majority of this work can be shared across branches, by developing reduction strategies that can be run parametrically on diagrams with boolean free parameters.As parameters only need to be fixed after the bulk of the simplification work is already done, we show that it is possible to perform the final stage of classical simulation quickly utilising a high degree of GPU parallelism.Using these methods, we demonstrate an average speedup factor of 78.3 ± 10.2 for certain classical simulation tasks vs. the non-parametric approach.

DOI
10.4204/eptcs.426.10

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It is known that a quantum circuit may be simulated with classical hardware via stabilizer state (T-)decomposition in [Formula: see text] time, given t non-Clifford gates and a decomposition efficiency [Formula: see text]. The past years have seen a number of papers presenting new decompositions of lower [Formula: see text] to reduce this runtime and enable simulation of ever larger circuits. More recently, it has been demonstrated that well-placed applications of apparently weaker (higher [Formula: see text]) decompositions can in fact result in better overall efficiency when paired with the circuit simplification strategies of ZX-calculus. In this work, we take the most generalized T-decomposition (namely vertex cutting), which achieves a poor efficiency of [Formula: see text] and identify common structures to which applying this can, after simplification via ZX-calculus rewriting, yield very strong effective efficiencies [Formula: see text]. By taking into account this broader scope of the ZX-diagram and incorporating the simplification facilitated by the well-motivated cuts, we derive a handful of efficient T-decompositions whose applicabilities are relatively frequent. In benchmarking these new ‘dynamic’ decompositions against the existing alternatives, we observe a significant reduction in overall [Formula: see text] and hence overall runtime for classical simulation, particularly for certain common circuit classes.

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The ZX-calculus is an algebraic formalism that allows quantum computations to be simplified via a small number of simple graphical rewrite rules.Recently, it was shown that, when combined with a family of "sum-over-Cliffords" techniques, the ZX-calculus provides a powerful tool for classical simulation of quantum circuits.However, for several important classical simulation tasks, such as computing the probabilities associated with many measurement outcomes of a single quantum circuit, this technique results in reductions over many very similar diagrams, where much of the same computational work is repeated.In this paper, we show that the majority of this work can be shared across branches, by developing reduction strategies that can be run parametrically on diagrams with boolean free parameters.As parameters only need to be fixed after the bulk of the simplification work is already done, we show that it is possible to perform the final stage of classical simulation quickly utilising a high degree of GPU parallelism.Using these methods, we demonstrate an average speedup factor of 78.3 ± 10.2 for certain classical simulation tasks vs. the non-parametric approach.
Abstract We introduce an enhanced technique for strong classical simulation of quantum circuits which combines the ‘sum-of-stabilisers’ method with an automated simplification strategy based on the ZX-calculus. Recently it was shown that quantum circuits can be classically simulated by expressing the non-stabiliser gates in a circuit as magic state injections and decomposing them in chunks of 2–6 states at a time, obtaining sums of (efficiently-simulable) stabiliser states with many fewer terms than the naive approach. We adapt these techniques from the original setting of Clifford circuits with magic state injection to generic ZX-diagrams and show that, by interleaving this ‘chunked’ decomposition with a ZX-calculus-based simplification strategy, we can obtain stabiliser decompositions that are many orders of magnitude smaller than existing approaches. We illustrate this technique to perform exact norm calculations (and hence strong simulation) on the outputs of random 50- and 100-qubit Clifford + T circuits with up to 70 T-gates as well as a family of hidden shift circuits previously considered by Bravyi and Gosset with over 1000 T-gates.
This article presents a novel algorithmic methodology for performing automated diagrammatic deductions over combinatorial structures, using a combination of modified equational theorem-proving techniques and the extended Wolfram model hypergraph rewriting formalism developed by the authors in previous work. We focus especially upon the application of this new algorithm to the problem of automated circuit simplification in quantum information theory, using Wolfram model multiway operator systems combined with the ZX-calculus formalism for enacting fast diagrammatic reasoning over linear transformations between qubits. We show how to construct a generalization of the deductive inference rules for Knuth-Bendix completion in which equation matches are selected on the basis of causal edge density in the associated multiway system, before proceeding to demonstrate how to embed the higher-order logic of the ZX-calculus rules within this first-order equational framework. After showing explicitly how the (hyper)graph rewritings of both Wolfram model systems and the ZX-calculus can be effectively realized within this formalism, we proceed to exhibit comparisons of time complexity vs. proof complexity for this new algorithmic approach when simplifying randomly-generated Clifford circuits down to pseudo-normal form, as well as when reducing the number of T-gates in randomly-generated non-Clifford circuits, with circuit sizes ranging up to 3000 gates, illustrating that the method performs favorably in comparison with existing circuit simplification frameworks, and also exhibiting the approximately quadratic speedup obtained by employing the causal edge density optimization. Finally, we present a worked example of an automated proof of correctness for a simple quantum teleportation protocol, in order to demonstrate more clearly the internal operations of the theorem-proving procedure.
Quantum computing is an emerging technology in which quantum mechanical properties are suitably utilized to perform certain compute-intensive operations faster than classical computers. Quantum algorithms are designed as a combination of quantum circuits that each require a large number of quantum gates, which is a challenge considering the limited number of qubit resources available in quantum computing systems. Our work proposes a technique to optimize quantum arithmetic algorithms by reducing the hardware resources and the number of qubits based on ZX calculus. We have utilised ZX calculus rewrite rules for the optimization of fault-tolerant quantum multiplier circuits where we are able to achieve a significant reduction in the number of ancilla bits and T-gates as compared to the originally required numbers to achieve fault-tolerance. Our work is the first step in the series of arithmetic circuit optimization using graphical rewrite tools and it paves the way for advancing the optimization of various complex quantum circuits and establishing the potential for new applications of the same.
In this paper, we introduce a technique for contracting (i.e. numerically evaluating) ZX-diagrams whose complexity scales with their rank-width, a graph parameter that behaves nicely under ZX rewrite rules. Given a rank-decomposition of width RRR, our method simulates a graph-like ZX-diagram in O~(4R)Õ(4^R)O~(4R) time. Applied to classical simulation of quantum circuits, it is no slower than either naive state vector simulation or stabiliser decompositions with α=0.5α= 0.5α=0.5, and in practice can be significantly faster for suitably chosen rank-decompositions. Since finding optimal rank-decompositions is NP-hard, we introduce heuristics that produce good decompositions in practice. We benchmark our simulation routine against Quimb, a popular tensor contraction library, and observe substantial reductions in floating-point operations (often by several orders of magnitude) for random and structured non-Clifford circuits as well as random ZX-diagrams.
The ZX-calculus is a graphical language for suitably represented tensor networks, called ZX-diagrams. Calculations are performed by transforming ZX-diagrams with rewrite rules. The ZX-calculus has found applications in reasoning about quantum circuits, condensed matter systems, quantum algorithms, quantum error correcting codes, and counting problems. A key notion is the stabiliser fragment of the ZX-calculus, a subfamily of ZX-diagrams for which rewriting can be done efficiently in terms of derived simplifying rewrites. Recently, higher dimensional qudits - in particular, qutrits - have gained prominence within quantum computing research. The main contribution of this work is the derivation of efficient rewrite strategies for the stabiliser fragment of the qutrit ZX-calculus. Notably, this constitutes a first non-trivial step towards the simplification of qutrit quantum circuits. We then give further unexpected areas in which these rewrite strategies provide complexity-theoretic insight; namely, we reinterpret known results about evaluating the Jones polynomial, an important link invariant in knot theory, and counting graph colourings.
It is known that a quantum circuit may be simulated with classical hardware via stabilizer state (T-)decomposition in [Formula: see text] time, given t non-Clifford gates and a decomposition efficiency [Formula: see text]. The past years have seen a number of papers presenting new decompositions of lower [Formula: see text] to reduce this runtime and enable simulation of ever larger circuits. More recently, it has been demonstrated that well-placed applications of apparently weaker (higher [Formula: see text]) decompositions can in fact result in better overall efficiency when paired with the circuit simplification strategies of ZX-calculus. In this work, we take the most generalized T-decomposition (namely vertex cutting), which achieves a poor efficiency of [Formula: see text] and identify common structures to which applying this can, after simplification via ZX-calculus rewriting, yield very strong effective efficiencies [Formula: see text]. By taking into account this broader scope of the ZX-diagram and incorporating the simplification facilitated by the well-motivated cuts, we derive a handful of efficient T-decompositions whose applicabilities are relatively frequent. In benchmarking these new ‘dynamic’ decompositions against the existing alternatives, we observe a significant reduction in overall [Formula: see text] and hence overall runtime for classical simulation, particularly for certain common circuit classes.