Useful fault-tolerant quantum computers will need a large supply of magic states: carefully prepared quantum resources that enable the operations ordinary error-corrected qubits cannot perform alone. Producing them efficiently is one of the major challenges on the path to practical quantum computing.
Magic state cultivation (MSC) is a recent family of recipes that produces magic states far more efficiently than older distillation methods. However, verifying how well these recipes actually work requires classically simulating circuits that fall outside the efficiently-simulable Clifford class, and the error rates of interest are as low as one in a billion.
In this work, we show that matrix product states (MPS) and Clifford-augmented MPS (CAMPS) — tensor network methods at the core of QPerfect’s MIMIQ™ platform — handle these circuits well, because the states involved stay only weakly entangled even when they are deeply non-Clifford. On the hardest circuit we studied, our CAMPS engine runs about 13× faster than the dedicated state-of-the-art near-Clifford simulator Clifft.
With this we settle two open questions. First, we propose a new cultivation scheme that reaches a logical error rate of 10−9 at roughly 1.3× lower space-time cost, using a recently introduced new computation protocol. Second, for fold-transversal surface-code cultivation (Sahay et al.), the cheap Clifford stand-in used by the community turns out to be a reliable proxy — unlike in the color-code case (Gidney et al.), where it is off by up to 30×.
Technical abstract
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 state (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 color-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.