1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, INF 229, D-69120 Heidelberg, Germany
3Imperial College London, London, UK
4Institute of Theoretical Physics, Jagiellonian University, Krakow, Poland.
5Instituto de Física Teórica, UAM/CSIC, Universidad Autónoma de Madrid, Madrid 28049, Spain
6Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
7Quantum Science Center, Oak Ridge, TN 37931, USA
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Abstract
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.
Featured image: On the left, there is a plot showing the average magnitude of the gradients of the circuit in the inset over circuit depth with different lines for the different numbers of dirty (noisy) and clean (noiseless) qubits in the simulated system. There is a clear exponential decay of the average magnitude of the gradients over circuit depth which is maximal when all qubits are noisy. On the right, there is a circuit diagram of how the noise channels $(mathcal N)$ are distributed around gates $(U)$ in the clean and dirty setup.
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