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arXiv:1811.09832

Consequences of unitary evolution of coupled qubit-resonator systems for stabilizing circuits in surface codes

H. W. L. Naus, R. Versluis

Published Nov 24, 2018• 2 citations

Abstract

Surface codes based on stabilizer circuits may pave the way for large scale fault-tolerant quantum computation. The surface code uses only single- and two-qubit gates and the error threshold falls close to 1% for a large range of errors. Among the most promising candidates to physically implement such circuits and codes are superconducting qubits coupled by resonators. We investigate a X and Z stabilizing circuit realized by two data qubits, two ancillas and four resonators. The aim is to assess the consequences of unitary evolution of the interacting system, in particular for given stable initial states, on fidelities and error syndrome probabilities. We model the system with a Jaynes-Tavis-Cummings Hamiltonian and construct the low-excitation level evolution operators. The analysis is limited to two stable input states. We assume an ideal system with perfect gates, perfect measurements and no decoherence or leakage. Our analysis shows that the capture probabilities after the execution of a single stabilizer round are not equal to 100%, but vary between 99.2% and 99.99%. This is caused solely by the unitary evolution of the interacting system. Two consecutive rounds of stabilizer measurements result in capture probability values that depend heavily on the duration of the evolution, but vary between 0% and 99%. Also due to the unitary evolution, the final state of the data qubits leaves the the four-dimensional subspace, which results in a state fidelity oscillating between 0 and 1. Even if an error on the qubits is captured, the correcting operation on the qubit will not bring the qubit to the original state. The errors induced by the Hamiltonian evolution of the system cannot be interpreted nor classified as commonly appearing errors. Additional or augmented quantum error correction may be required to compensate these effects of resonator-qubit interaction.

DOI: 10.48550/arxiv.1811.09832

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Surface codes based on stabilizer circuits may pave the way for large scale fault-tolerant quantum computation. The surface code uses only single- and two-qubit gates and the error threshold falls close to 1% for a large range of errors. Among the most promising candidates to physically implement such circuits and codes are superconducting qubits coupled by resonators. We investigate a X and Z stabilizing circuit realized by two data qubits, two ancillas and four resonators. The aim is to assess the consequences of unitary evolution of the interacting system, in particular for given stable initial states, on fidelities and error syndrome probabilities. We model the system with a Jaynes-Tavis-Cummings Hamiltonian and construct the low-excitation level evolution operators. The analysis is limited to two stable input states. We assume an ideal system with perfect gates, perfect measurements and no decoherence or leakage. Our analysis shows that the capture probabilities after the execution of a single stabilizer round are not equal to 100%, but vary between 99.2% and 99.99%. This is caused solely by the unitary evolution of the interacting system. Two consecutive rounds of stabilizer measurements result in capture probability values that depend heavily on the duration of the evolution, but vary between 0% and 99%. Also due to the unitary evolution, the final state of the data qubits leaves the the four-dimensional subspace, which results in a state fidelity oscillating between 0 and 1. Even if an error on the qubits is captured, the correcting operation on the qubit will not bring the qubit to the original state. The errors induced by the Hamiltonian evolution of the system cannot be interpreted nor classified as commonly appearing errors. Additional or augmented quantum error correction may be required to compensate these effects of resonator-qubit interaction.

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Superconducting qubits, while promising for scalability and long coherence times, contain more than two energy levels, and therefore are susceptible to errors generated by the leakage of population outside of the computational subspace. Such leakage errors constitute a prominent roadblock towards fault-tolerant quantum computing (FTQC) with superconducting qubits. FTQC using topological codes is based on sequential measurements of multiqubit stabilizer operators. Here, we first propose a leakage-resilient procedure to perform repetitive measurements of multiqubit stabilizer operators, and then use this scheme as an ingredient to develop a leakage-resilient approach for surface code quantum error correction with superconducting circuits. Our protocol is based on swap operations between data and ancilla qubits at the end of every cycle, requiring read-out and reset operations on every physical qubit in the system, and thereby preventing persistent leakage errors from occurring.

Four electron spin qubits in quantum dots are studied by means of an exchange interaction Hamiltonian. The time-independent Schrödinger equation is exactly analytically solved for the symmetric case, that is equal qubit frequencies and equal couplings. The spin system can be used as two data-qubits and two ancillas, thereby realizing a

X

and

Z

stabilizing circuit exploited in error correction and surface codes. The unitary evolution of corresponding stable states is calculated and consequences for error syndromes and fidelities are assessed. It is also shown that applying the evolution operator for a prescribed duration and subsequent measurement of the ancillas lead to a `verifiable' two-qubit phase gate.

Quantum error correction (QEC) represents a critical enabler for fault-tolerant quantum computing, protecting fragile quantum states from decoherence and operational errors. This paper provides a comprehensive analysis of leading quantum error correction schemes, focusing on surface codes and topological codes within the context of Noisy Intermediate-Scale Quantum (NISQ) era processors. We examine the theoretical foundations of stabilizer codes, compare surface code implementations with code distances ranging from d=3 to d=17, and analyze topological codes including toric codes and color codes. Our investigation encompasses syndrome extraction circuits, logical qubit encoding strategies, and threshold analysis for contemporary quantum hardware platforms. Performance metrics reveal that surface codes achieve error thresholds of approximately 1% with realistic noise models, while topological codes offer architectural advantages for specific qubit connectivity constraints. We discuss implementation challenges on current NISQ processors including IBM Quantum, Google Sycamore, and IonQ systems, addressing qubit overhead, gate fidelity requirements, and decoding algorithms. This analysis provides insights for selecting appropriate error correction schemes based on application requirements and hardware capabilities.

The construction of topological error correction codes requires the ability to fabricate a lattice of physical qubits embedded on a manifold with a nontrivial topology such that the quantum information is encoded in the global degrees of freedom (i.e., the topology) of the manifold. However, the manufacturing of large-scale topological devices will undoubtedly suffer from fabrication errors---permanent faulty components such as missing physical qubits or failed entangling gates---introducing permanent defects into the topology of the lattice and hence significantly reducing the distance of the code and the quality of the encoded logical qubits. In this work we investigate how fabrication errors affect the performance of topological codes, using the surface code as the test bed. A known approach to mitigate defective lattices involves the use of primitive swap gates in a long sequence of syndrome extraction circuits. Instead, we show that in the presence of fabrication errors the syndrome can be determined using the supercheck operator approach and the outcome of the defective gauge stabilizer generators without any additional computational overhead or use of swap gates. We report numerical fault-tolerance thresholds in the presence of both qubit fabrication and gate fabrication errors using a circuit-based noise model and the minimum-weight perfect-matching decoder. Our numerical analysis is most applicable to two-dimensional chip-based technologies, but the techniques presented here can be readily extended to other topological architectures. We find that in the presence of

8%

qubit fabrication errors, the surface code can still tolerate a computational error rate of up to

0.1%

.

Recently, significant effort has been devoted towards designing erasure qubits in which dominant physical noise excites detectable leakage states that can be returned to the qubit subspace. Interest in these erasure qubits has been driven by studies showing that the requirements for fault-tolerant quantum error correction are significantly relaxed when noise in every gate operation is dominated by erasures. However, these studies assume perfectly accurate erasure checks after every gate operation which generally come with undesirable time and hardware overhead costs. In this work, we investigate the consequences of using an imperfect but overhead-efficient erasure check for fault-tolerant quantum error correction with the surface code. We show that, under physically reasonable assumptions on the imperfect erasure checks, the threshold error rate is still at least over twice that for Pauli noise. We also study the impact of imperfect erasure checks on the effective error distance and find that it degrades the effective distance under a general error model in which a qubit suffers from depolarizing noise when interacting with a leaked qubit. We then identify a more restrictive but realistic noise model for a qubit that interacts with a leaked qubit, under which the effective error distance is twice that for Pauli noise. We apply our analysis to recently proposed superconducting dual-rail erasure qubits and show that achieving good-performance surface code quantum memories with relaxed system requirements is possible.