Simulating quantum error correction (QEC) circuits including non-Clifford gates at scale is important to accelerate progress toward fault-tolerant quantum computing. Here we demonstrate that matrix product state (MPS) techniques can handle many QEC circuits exactly and without restriction on gate types. Crucially, we find that MPS efficiency depends sensitively on implementation choices, and we introduce a series of targeted optimizations that reduce bond dimensions and simulation time by several orders of magnitude compared to naive approaches. We illustrate this with examples including: (a) a rotated surface code quantum memory up to distance 11, (b) logical Bell-state preparation up to distance 9, (c) a 15-to-1 magic-state distillation circuit including hundreds of QEC rounds that we optimize to be simulated with only 11 logical qubits (187 physical qubits) and a maximal bond dimension of 64 in under 40 seconds, and (d) a narrow, deep random circuit that scales linearly with the number of T gates. These results demonstrate the importance of circuit-level optimizations and position MPS as a valuable complement to near-Clifford simulators for QEC circuits.
Simulating quantum error correction (QEC) circuits including non-Clifford gates at scale is important to accelerate progress toward fault-tolerant quantum computing. Here we demonstrate that matrix product state (MPS) techniques can handle many QEC circuits exactly and without restriction on gate types. Crucially, we find that MPS efficiency depends sensitively on implementation choices, and we introduce a series of targeted optimizations that reduce bond dimensions and simulation time by several orders of magnitude compared to naive approaches. We illustrate this with examples including: (a) a rotated surface code quantum memory up to distance 11, (b) logical Bell-state preparation up to distance 9, (c) a 15-to-1 magic-state distillation circuit including hundreds of QEC rounds that we optimize to be simulated with only 11 logical qubits (187 physical qubits) and a maximal bond dimension of 64 in under 40 seconds, and (d) a narrow, deep random circuit that scales linearly with the number of T gates. These results demonstrate the importance of circuit-level optimizations and position MPS as a valuable complement to near-Clifford simulators for QEC circuits.
Magic state cultivation prepares high-fidelity magic states at low expected space-time costs; however, the exact performance of some schemes is unsettled due to the difficulty in simulating non-Clifford circuits. Here, we use matrix-product states (MPS) based methods to compute the exact performance of two types of fold-transversal cultivation schemes: (i) the Sahay et al method based on the regular surface code S gate, and (ii) a method we propose based on a partially fault-tolerant fold-transversal S gate. We show that for the former protocol at d=5, the ∣T⟩ output reaches similar logical error rates to the ∣S⟩ output, traditionally used as a cheap full Clifford proxy. This contrasts with the ∼10× discrepancy reported for the d=5 colour-code scheme of Gidney et al. We also find that our new d=5 scheme has ∼1.3× lower expected space-time cost while still reaching 10−9 logical error rate. We show that MPS and Clifford-augmented MPS (CAMPS) perform on par with or even better than the recently introduced near-Clifford simulator Clifft on the hardest d=5 regular surface code scheme. Additionally, to speed up simulation, we propose a new pre-screening method based on simple Pauli propagation, lowering by up to three orders of magnitude the required number of exact simulations, and use several simulator-agnostic sampling methods such as subset sampling.
Quantum Error Correction (QEC), combined with magic state distillation, ensures fault tolerance in large-scale quantum computation. To apply QEC, a circuit must first be transformed into a non-Clifford (or T) gate set. T-depth, the number of sequential T-gate layers, determines the magic state cost, impacting both spatial and temporal overhead. Minimizing T-depth is crucial for optimizing resource efficiency in fault-tolerant quantum computing. While QEC scalability has been widely studied, T-depth reduction remains an overlooked challenge. It is established that T-depth reduction is an NP-hard problem and we systematically evaluate multiple approximation techniques: greedy, divide-and-conquer, Lookahead-based brute force, and graph-based. The Lookahead-based brute-force algorithm (partition size 4) performs best, optimizing 90% of reducible cases (i.e., circuits where at least one algorithm achieved optimization) with an average T-depth reduction of around 51%. Additionally, we introduce an expansion factor-based identity gate insertion strategy, leveraging controlled redundancy to achieve deeper reductions in circuits initially classified as non-reducible. With this approach, we successfully convert up to 25% of non-reducible circuits into reducible ones, while achieving an additional average reduction of up to 11.8%. Furthermore, we analyze the impact of different expansion factor values and explore how varying the partition size in the Lookahead-based brute-force algorithm influences the quality of T-depth reduction.
We introduce a family of quantum circuits that possess standard indicators of classical simulation hardness including high entanglement entropy, magic, and non-Gaussianity, yet admit efficient classical simulation via matrix product states (MPS). Our construction uses logical circuits of high-rate Calderbank-Shor-Steane (CSS) codes with enhanced symmetries. Using code automorphisms and transversal diagonal gates from higher levels of the Clifford hierarchy, we realize nonlocal logical Clifford and non-Clifford gates, showing how error-correcting codes can compile complex logical circuits into simple physical operations. Simulation efficiency rests on two properties: (i) diagonal transversal gates do not increase bond dimension, and (ii) permutations are tracked classically via on-the-fly relabeling, avoiding costly SWAP networks. Unlike Clifford or matchgate simulation, our method accepts a broad class of initial states, including dense entangled, magic, and non-Gaussian inputs, provided the encoded state retains an efficient MPS representation. We also release an exact phase-polynomial backend for monomial subfamilies, whose cost is set by higher-degree phase terms rather than entanglement growth. We demonstrate the method on an infinite polar CSS code family, showing bond dimension stays bounded by the encoding cost regardless of circuit depth. These results show that for some circuit families, standard resource measures are individually insufficient to indicate simulation hardness. As a near-term application, we use the compiled MPS as a classical reference for direct fidelity estimation of a quantum device running nontrivial logical circuits. Pauli sampling on the encoded reference, with a Clifford pushback through the known encoder, provides the ideal expectation values, so the logical output fidelity can be estimated from local Pauli readout alone, without costly state tomography.
The classical simulation of universal quantum circuits is crucial both fundamentally and practically for quantum computation. We propose SyQMA, a simulator with several convenient features, particularly suited for quantum error correction (QEC). SyQMA simulates universal quantum circuits with incoherent Pauli noise and computes exact expectation values and measurement probabilities as symbolic functions of circuit parameters: rotation angles, measurement outcomes, and noise rates. This simulator can sample measurement outcomes, enabling the simulation of dynamic quantum programs where circuit composition depends on prior measurement outputs. For QEC, it performs circuit-level maximum-likelihood decoding, provides exact symbolic expressions for logical error rates, and verifies the fault distance of fault-tolerant (FT) stabiliser and magic state preparation protocols. These features are enabled by an intuitive extension of stabiliser simulators, where each non-Clifford Pauli rotation and incoherent Pauli channel is compactly represented via auxiliary qubits and a modified trace. Representing the state requires only polynomial memory and time, while computing expectation values and measurement probabilities takes exponential time in the number of non-Clifford rotations and deterministic measurements, but only polynomial memory. The FT preparation of stabiliser and magic states, including the first stage of magic state cultivation, is analysed without approximations. We also exactly convert the disjoint error probabilities of a general multi-qubit Pauli channel to independent ones, a key step for creating and sampling from detector error models. The code is publicly available and open-source.
Quantum error correction (QEC), the lynchpin of fault-tolerant quantum computing (FTQC), is designed and validated against well-behaved Pauli stochastic error models. But in real-world deployment, QEC protocols encounter a vast array of other errors -- coherent and non-Pauli errors -- whose impacts on quantum circuits are vastly different than those of stochastic Pauli errors. The impacts of these errors on QEC and FTQC protocols have been largely unpredictable to date due to exponential classical simulation cost. Here, we show how to accurately and efficiently model the effects of coherent and non-Pauli errors on FTQC, and we study the effects of such errors on syndrome extraction for surface and bivariate bicycle codes, and on magic state cultivation. Our analysis suggests that coherent error can shift fault-tolerance thresholds, increase the space-time cost of magic state cultivation, and can increase logical error rates by an order of magnitude compared to equivalent stochastic errors. These analyses are enabled by a new technique for mapping any Markovian circuit-level error model with sufficiently small error rates onto a detector error model (DEM) for an FTQC circuit. The resulting DEM enables Monte Carlo estimation of logical error rates and noise-adapted decoding, and its parameters can be analytically related to the underlying physical noise parameters to enable approximate strong simulation.
Zero-level CCZ Distillation
Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the T gate and the CCZ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical CCZ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal T/T† operation of the [[8,3,2]] code to fault-tolerantly encode the state CCZ∣+++⟩, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as pL≃300×p2 with respect to the physical error rate p. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at p=10−3 and p=10−4, respectively, compared to conventional seven-T-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of CCZ-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional T-state generation.