We study error correction type protocols in which a quantum channel encodes logical information into an enlarged Hilbert space. Specifically, we consider channels realized by one dimensional random noisy quantum circuits with spatially local interaction gates. We analyze both noise acting after the encoding and noise affecting the encoding circuit itself. Using the coherent information as a metric, we show that in both cases the infinite depth limit is governed by random matrix theory, which predicts a universal phase transition at a critical noise rate. This critical point separates an error correcting phase, in which encoded information is preserved, from a phase in which it is irretrievably lost. Going beyond the infinite depth limit, we characterize the systematic finite depth deviations from random matrix universality. In particular, we show that these deviations behave parametrically differently depending on whether the noise acts after the encoding or also affects the encoding itself. For noiseless encoders, the approach is exponential in circuit depth, although boundary effects can delay perfect encoding relative to the circuit design time. For noisy encoders, we find that the circuit fidelity effectively replaces the Hashing bound, and perfect encoding is approached polynomially with depth.
In this work, we study the task of encoding logical information via a noisy quantum circuit. It is known that at superlogarithmic depth, the output of any noisy circuit without reset gates or intermediate measurements becomes indistinguishable from the maximally mixed state, implying that all input information is destroyed. This raises the question of whether there is a low-depth regime where information is preserved as it is encoded into an error-correcting codespace by the circuit. When considering noisy random encoding circuits, our numerical simulations show that there is a sharp phase transition at a critical depth of order
p−1, where
p is the noise rate, such that below this depth threshold quantum information is preserved, whereas after this threshold it is lost. Furthermore, we rigorously prove that this is the best achievable trade-off between depth and noise rate for any noisy circuit encoding a constant rate of information. Thus, random circuits are optimal noisy encoders in this sense.
Random unitaries scramble quantum information into highly nonlocal degrees of freedom, thereby providing protection from local noise. However, as demonstrated by Turkeshi and Sierant (Phys. Rev. Lett. 132, 140401, 2024 and arXiv:2603.00235), this protection may fail once local noise grows sufficiently strong. These works describe a phase transition in a noisy random encoding-decoding circuit model between an error-protecting phase, in which encoded information is recoverable, and an error-vulnerable phase, in which encoded information is lost, with the transition occurring at a critical coherent noise angle
αc≈1.14. Here, we present the first experimental observation of this transition, conducted on IBM's ibm_pittsburgh Heron r3 device for system sizes of up to
N=26 qubits. We detect this transition via three order parameters: fidelity, Rényi-2 entanglement entropy, and stabilizer Rényi-2 entropy. For the latter two, we employ randomized measurement protocols to construct estimators, and for all three, we rely on zero-noise extrapolation and M3 SPAM error mitigation. We then jointly model the finite-size scaling collapse across all three order parameters with a Gaussian process regression, yielding a critical angle
αc≈1.01, well below the theoretical value. Inputting noiseless simulated data into the same model yields
αc≈1.05, implying that the discrepancy between experiment and theory is due to our encoding circuits, which are not deep enough to be truly Haar-random. Finally, we observe that the same model cannot adequately identify critical behavior in unmitigated, raw experimental data. This indicates that error mitigation techniques are crucial for detecting the transition on hardware.
We present universal properties of anticoncentration in noisy quantum circuits at finite depth. We develop a generic framework for single- and multi-qubit noise channels in the weak-noise limit and introduce an effective description in terms of a . Within this weak-noise regime, we show that distinct noise mechanisms act in a quantitatively similar way, yielding a universal distribution of bit-string probabilities that is largely independent of the microscopic noise channel and of the circuit architecture. We identify three depth-dependent regimes, each characterized by a distinct scaling of cross-entropy benchmarking ( XEB ) with rescaled depth. In the shallow-depth regime, noise effects are perturbatively small; in the intermediate regime, circuit-induced fluctuations and noise compete on equal footing; and in the deep-depth regime, the output distribution becomes effectively classical, up to corrections that are exponentially small in the noise strength. We provide quantitative predictions for anticoncentration in generic finite-depth circuits and benchmark them against numerical simulations, finding excellent agreement even at shallow depths. Moreover, we show that, contrary to previous expectations, the late-time value of XEB provides direct access to the global circuit fidelity, even at large noise strengths. Our results are directly applicable to current quantum processors and demonstrate universal behavior beyond the pure random-matrix-theory regime, which only emerges at asymptotically large depths.
Efficient and high-performance quantum error correction is essential for achieving fault-tolerant quantum computing. Low-depth random circuits offer a promising approach to identifying effective and practical encoding strategies. In this work, we rigorously prove through information-theoretic analysis that one-dimensional logarithmic-depth random Clifford encoding circuits can achieve high quantum error correction performance. We demonstrate that these random codes typically exhibit good approximate quantum error correction capability by proving that their encoding rate achieves the hashing bound for Pauli noise and the channel capacity for erasure errors. We show that the error correction inaccuracy decays once a threshold of logarithmic depth is exceeded, resulting in negligible recovery errors. This threshold is shown to be lower than that of the simple separate block encoding, and the decay rate is higher. We further establish that these codes are optimal by proving that logarithmic depth is necessary to maintain a constant encoding rate and high error correction performance. To prove our results, we propose decoupling theorems tailored for one-dimensional low-depth circuits. These results also imply strong decoupling and rapid thermalization properties in low-depth random circuits and have potential applications in quantum information science and physics.
Abstract Quantum deviations or coherent noise are a typical type of noise when implementing gate operations in quantum computers, and their impact on the performance of quantum error correction (QEC) is still elusive. Here, we consider the topological surface code, with both stochastic noise and coherent noise on the multi-qubit entanglement gates during stabilizer measurements in both initial state preparation and error detection. We map a multi-round error detection protocol to a three-dimensional statistical mechanical model consisting of Z_2 gauge interactions and related the error threshold to its phase transition point. Specifically, two error thresholds are identified distinguishing different error correction performances. Below a finite error threshold, in stark contrast to the case with only stochastic errors, unidentifiable measurement errors can cause the failure of QEC in the large code distance limit. This problem can only be fixed at the perfect initial state preparation point. For a finite or small code with distance d, we find that if the preparation error rate is below a crossover scale ~1/log(d), the logical errors can still be suppressed. We conclude that this type of unavoidable coherent noise has a significant impact on QEC performance, and becomes increasingly detrimental as the code distance increases.
A quantum error-correcting code with a nonzero error threshold undergoes a mixed-state phase transition when the error rate reaches that threshold. We explore this phase transition for Haar-random quantum codes, in which the logical information is encoded in a random subspace of the physical Hilbert space. We focus on the spectrum of the encoded system density matrix as a function of the rate of uncorrelated, single-qudit errors. For low error rates, this spectrum consists of well-separated bands, representing errors of different weights. As the error rate increases, the bands for high-weight errors merge. The evolution of these bands with increasing error rate is well described by a simple analytic ansatz. Using this ansatz, as well as an explicit calculation, we show that the threshold for Haar-random quantum codes saturates the hashing bound, and thus coincides with that for random codes. For error rates that exceed the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher threshold. Postselection can in principle be implemented by projecting onto subspaces corresponding to low-weight errors, which remain correctable past the hashing bound.