Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of s t a b i l i z e r rank , which for a pure state ψ is defined to be the smallest integer χ such that ψ is a superposition of χ stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of χ ∼ 10 6 stabilizer states and sample from the full n -bit output distribution, improving on previous simulations which used ∼ 10 3 stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64 T gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.
Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of \ns\nt\na\nb\ni\nl\ni\nz\ne\nr\n \nrank\n, which for a pure state \nψ\n is defined to be the smallest integer \nχ\n such that \nψ\n is a superposition of \nχ\n stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of \nχ\n∼\n10\n6\n stabilizer states and sample from the full \nn\n-bit output distribution, improving on previous simulations which used \n∼\n10\n3\n stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64 \nT\n gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.
We present an exact amplitude simulator for Clifford+T quantum circuits that combines a Feynman path sum across a balanced qubit bipartition with stabilizer-frame simulation on each half. The construction extends prior stabilizer-based Schrödinger-Feynman methods in three directions: recursive multilevel bipartition into a binary tree, automatic fallback to dense state-vector simulation when a leaf's stabilizer frame would exceed its memory ceiling, and a cost-model-driven partition selector that replaces the standard cut-count minimization heuristic.
We show cut-count minimization is an unreliable proxy in practice: a globally cleaner partition can reduce cross-cut count yet increase wall-clock time, because it imbalances T-gate density across halves and inflates per-half stabilizer-frame size. Our cost model substitutes the stabilizer-frame bound 2w for the dense 2n ceiling per side and explicitly models per-amplitude readout cost; isolating that term uncovered a quadratic-asymptotic inefficiency in the leaf simulator's end-of-path amplitude extraction, fixed by replacing it with an existing O(F * s * n) single-amplitude inner product.
On a structured hierarchical n=16 benchmark the recursive simulator beats monolithic stabilizer-frame simulation by 92x to 17,645x, wins by 79x per path against a dense half-state-vector baseline under an identical cut, and beats a production state-vector simulator end to end by up to 47.9x (median ~5x). On adversarial random Clifford+T circuits the dense state vector wins past a crossover near n/2 cross-cut gates -- the regime the cost model identifies. The dominant cost, the cross-cut Feynman sum, is embarrassingly parallel with constant inter-worker communication, unlike recent matrix-product-state stabilizer-tensor methods whose inner contraction loop is sequential.
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.
Despite the widespread belief that quantum computers cannot be efficiently simulated classically, efficient simulation is known to be possible in certain restricted regimes. In particular, the Gottesman-Knill theorem states that Clifford circuits can be efficiently simulated. We begin this thesis by reviewing and comparing several known techniques for efficient simulation of Clifford circuits: the stabilizer formalism, CH form, affine form, and the graph state formalism. We describe each simulation method and give four different proofs of the Gottesman-Knill theorem. \n \nNext we review a recent work [15], which shows that restricting the geometry of Clifford circuits can lead to a further speedup. We give an algorithm for simulating Pauli basis measurements on a planar graph state in time
widetildeO(nomega/2), where
omega<2.373 is the matrix multiplication exponent. This algorithm achieves a quadratic speedup over using Clifford simulation methods directly. As an application of this algorithm, we consider a depth-
d Clifford circuit whose two-qubit gates act along edges of a planar graph and describe how to sample from its output distribution or compute an output probability in time $\\widetilde{O}(n^{\\omega/2}d^\\omega)$. For
d=O(logn), both of these results are quadratic speedups over using Clifford simulation methods directly. \n \nFinally, we extend these simulation algorithms to universal circuits by using stabilizer rank methods. We follow a previously known gadgetization procedure [9] to show that given a depth-
d Clifford+
T circuit with
t T gates and whose two-qubit gates act along edges of a planar graph, we can sample from its output distribution in time
widetildeO(20.7926tn5/2t6d3) and can compute output probabilities in time
widetildeO(20.3963tn3/2t6d3). Previous work [9,6], applied to the case
d=O(logn), gives algorithms for sampling in time
O(20.3963tn6t6) and computation of output probabilities in time
O(20.3963tn3t3). Our sampling algorithm offers improved scaling in
n but poorer scaling in the exponential term, while our algorithm for computing output probabilities offers improved scaling in
n with identical scaling in the exponential term.
One of the state of the art techniques for classically simulating quantum circuits relies on approximating the output state of the circuit by a superposition of stabilizer states. \nIf the number of non-Clifford gates in the circuit is small, such simulations can be very effective. \nThis thesis provides various improvements in this framework. First, we describe an improved method of computing approximate stabilizer decompositions, which reduces the time cost of computing a single term in the decomposition from
O(elln2) to
O(mn2), where
ell is the total number of gates in the circuit, and
m is the number of non-Clifford gates. \nSince this subroutine has to be repeated exponentially many times, this improvement can be significant in practice whenever
ellggm. \nOur method uses a certain re-writing of the circuit, which in some cases allows for a significant amelioration of the exponential scaling of the required classical resources. \n \nFurthermore, we describe a method of constructing exact, low-rank stabilizer decompositions of
ketpsiotimesm, where
ketpsi is either a magic state or an equatorial state. \nFor any single qubit magic state
ketpsi, we find stabilizer decompositions of
ketpsiotimesm with
2mlog2(3)/4 terms. This improves on the best known bound of
2mlog2(7)/6. \nSimilarly, for any single qubit equatorial state
ketpsi, we give a stabilizer decomposition of
ketpsiotimesm with
2m/2 terms. \nTo our knowledge no such decompositions were previously known. \nThese results translate to milder exponential scaling of the classical resources required for estimating probabilities of quantum circuits up to a polynomially small multiplicative error, as well as allowing more types of circuits to be simulated in this way. \n \nWe also consider certain obstructions to classical simulations. \nIt has been argued in various contexts that contextuality and non-locality hamper classical simulations of quantum circuits. \nLinear constraint systems (LCSs) are a generalization of the well-known Peres-Mermin magic square, which has been recently used to prove a separation between the power of constant depth classical and quantum circuits. \n \nWhile binary LCSs have been studied in detail,
d-ary LCSs are less well-understood. In this thesis we consider linear constraint systems modulo
d>2. We give a simple proof, of the previously known fact, that any linear constraint system which admits a quantum solution consisting of generalized Pauli observables in odd dimension must be classically satisfiable. We further prove that, for odd
d, if a Pauli-like commutation relation between two variables in the LCS arises, then it has no quantum solutions in any dimensions, in stark contrast to the even
d case. We apply this result to various examples, for instance showing that many generalizations of the Peres-Mermin magic square do not give rise to a quantum vs. classical satisfiability gap.