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As quantum computing hardware steadily increases in qubit count and quality, one important question is how to allocate these resources to mitigate the effects of hardware noise. In a transitional era between noisy small-scale and fully fault-tolerant systems, we envisage a scenario in which we are only able to error correct a small fraction of the qubits required to perform an interesting computation. In this work, we develop concrete constructions of logical operations on a joint system of a collection of noisy and a collection of error-corrected logical qubits. Within this setting and under Pauli noise assumptions, we provide analytical evidence that brick-layered circuits display on average slower concentration to the "useless" uniform distribution with increasing circuit depth compared to fully noisy circuits. We corroborate these findings by numerical demonstration of slower decoherence with an increasing fraction of error-corrected qubits under a noise model inspired by a real device. We find that this advantage only comes when the number of error-corrected qubits passes a specified threshold which depends on the number of couplings between error-corrected and noisy registers., Comment: 14+9 pages, 7+2 figures, comments welcome

Mixed quantum-classical dynamics is a set of methods often used to understand systems too complex to treat fully quantum mechanically. Many techniques exist for full quantum mechanical evolution on quantum computers, but mixed quantum-classical dynamics are less explored. We present a modular algorithm for general mixed quantum-classical dynamics where the quantum subsystem is coupled with the classical subsystem. We test it on a modified Shin-Metiu model in the first quantization through Ehrenfest propagation. We find that the Time-Dependent Variational Time Propagation algorithm performs well for short-time evolutions and retains qualitative results for longer-time evolutions., Comment: 13 pages, 19 figures, revision after first referee round

When error correction becomes possible it will be necessary to dedicate a large number of physical qubits to each logical qubit. Error correction allows for deeper circuits to be run, but each additional physical qubit can potentially contribute an exponential increase in computational space, so there is a trade-off between using qubits for error correction or using them as noisy qubits. In this work we look at the effects of using noisy qubits in conjunction with noiseless qubits (an idealized model for error-corrected qubits), which we call the "clean and dirty" setup. We employ analytical models and numerical simulations to characterize this setup. Numerically we show the appearance of Noise-Induced Barren Plateaus (NIBPs), i.e., an exponential concentration of observables caused by noise, in an Ising model Hamiltonian variational ansatz circuit. We observe this even if only a single qubit is noisy and given a deep enough circuit, suggesting that NIBPs cannot be fully overcome simply by error-correcting a subset of the qubits. On the positive side, we find that for every noiseless qubit in the circuit, there is an exponential suppression in concentration of gradient observables, showing the benefit of partial error correction. Finally, our analytical models corroborate these findings by showing that observables concentrate with a scaling in the exponent related to the ratio of dirty-to-total qubits., Comment: 27 pages, 15 figures, (v2) minor changes

Error mitigation is an essential component of achieving a practical quantum advantage in the near term, and a number of different approaches have been proposed. In this work, we recognize that many state-of-the-art error mitigation methods share a common feature: they are data-driven, employing classical data obtained from runs of different quantum circuits. For example, Zero-noise extrapolation (ZNE) uses variable noise data and Clifford-data regression (CDR) uses data from near-Clifford circuits. We show that Virtual Distillation (VD) can be viewed in a similar manner by considering classical data produced from different numbers of state preparations. Observing this fact allows us to unify these three methods under a general data-driven error mitigation framework that we call UNIfied Technique for Error mitigation with Data (UNITED). In certain situations, we find that our UNITED method can outperform the individual methods (i.e., the whole is better than the individual parts). Specifically, we employ a realistic noise model obtained from a trapped ion quantum computer to benchmark UNITED, as well as other state-of-the-art methods, in mitigating observables produced from random quantum circuits and the Quantum Alternating Operator Ansatz (QAOA) applied to Max-Cut problems with various numbers of qubits, circuit depths and total numbers of shots. We find that the performance of different techniques depends strongly on shot budgets, with more powerful methods requiring more shots for optimal performance. For our largest considered shot budget ($10^{10}$), we find that UNITED gives the most accurate mitigation. Hence, our work represents a benchmarking of current error mitigation methods and provides a guide for the regimes when certain methods are most useful., Comment: 25 pages, 11 figures, extended theoretical and numerical results, accepted by Quantum

We report the observation and characterisation of a systematic error in the implementation of $U_3$ gates in the IBM quantum computers. The error appears as a consistent shift in one of the angles of the gate, whose magnitude does not correlate with IBM's cited errors calculated using Clifford randomized benchmarking. We propose a simple mitigation procedure, leading to an improvement in the observed value for the CHSH inequality, highlighting the utility of simple mitigation strategies for short-depth quantum circuits., Comment: 9 pages, 8 figures