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arXiv:2201.00360 · Physical Review Research

Algebraic structure of path-independent quantum control

Wen-Long Ma, Shu‐Shen Li, Liang Jiang

Published May 9, 2022• 6 citations
arXivHTMLPDF

Abstract

Path-independent (PI) quantum control has recently been proposed to integrate quantum error correction and quantum control [W.-L. Ma, M. Zhang, Y. Wong, K. Noh, S. Rosenblum, P. Reinhold, R. J. Schoelkopf, and L. Jiang, Phys. Rev. Lett. 125, 110503 (2020)], achieving fault-tolerant quantum gates against ancilla errors. Here we reveal the underlying algebraic structure of PI quantum control. The PI Hamiltonians and propagators turn out to lie in an algebra isomorphic to the ordinary matrix algebra, which we call the PI matrix algebra. The PI matrix algebra, defined on the Hilbert space of a composite system (including an ancilla system and a central system), is isomorphic to the matrix algebra defined on the Hilbert space of the ancilla system. By extending the PI matrix algebra to the Hilbert-Schmidt space of the composite system, we provide an exact and unifying condition for PI quantum control against ancilla noise.

DOI
10.1103/physrevresearch.4.023102

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Path-independent (PI) quantum control has recently been proposed to integrate quantum error correction and quantum control [W.-L. Ma, M. Zhang, Y. Wong, K. Noh, S. Rosenblum, P. Reinhold, R. J. Schoelkopf, and L. Jiang, Phys. Rev. Lett. 125, 110503 (2020)], achieving fault-tolerant quantum gates against ancilla errors. Here we reveal the underlying algebraic structure of PI quantum control. The PI Hamiltonians and propagators turn out to lie in an algebra isomorphic to the ordinary matrix algebra, which we call the PI matrix algebra. The PI matrix algebra, defined on the Hilbert space of a composite system (including an ancilla system and a central system), is isomorphic to the matrix algebra defined on the Hilbert space of the ancilla system. By extending the PI matrix algebra to the Hilbert-Schmidt space of the composite system, we provide an exact and unifying condition for PI quantum control against ancilla noise.
Ancilla systems are often indispensable to universal control of a nearly isolated quantum system. However, ancilla systems are typically more vulnerable to environmental noise, which limits the performance of such ancilla-assisted quantum control. To address this challenge of ancilla-induced decoherence, we propose a general framework that integrates quantum control and quantum error correction, so that we can achieve robust quantum gates resilient to ancilla noise. We introduce the path independence criterion for fault-tolerant quantum gates against ancilla errors. As an example, a path-independent gate is provided for superconducting circuits with a hardware-efficient design.
Fault-tolerant quantum computation with bosonic qubits often necessitates the use of noisy discrete-variable ancillae. In this work, we establish a comprehensive and practical fault-tolerance framework for such a hybrid system and synthesize it with fault-tolerant protocols by combining bosonic quantum error correction (QEC) and advanced quantum control techniques. We introduce essential building blocks of error-corrected gadgets by leveraging ancilla-assisted bosonic operations using a generalized variant of path-independent quantum control. Using these building blocks, we construct a universal set of error-corrected gadgets that tolerate a single-photon loss and an arbitrary ancilla fault for four-legged cat qubits. Notably, our construction requires only dispersive coupling between bosonic modes and ancillae, as well as beam-splitter coupling between bosonic modes, both of which have been experimentally demonstrated with strong strengths and high accuracy. Moreover, each error-corrected bosonic qubit is comprised of only a single bosonic mode and a three-level ancilla, featuring the hardware efficiency of bosonic QEC in the full fault-tolerant setting. We numerically demonstrate the feasibility of our schemes using current experimental parameters in the circuit-QED platform. Finally, we present a hardware-efficient architecture for fault-tolerant quantum computing by concatenating the four-legged cat qubits with an outer qubit code utilizing only beam-splitter couplings. Our estimates suggest that the overall noise threshold can be reached using existing hardware. These developed fault-tolerant schemes extend beyond their applicability to four-legged cat qubits and can be adapted for other rotation-symmetrical codes, offering a promising avenue toward scalable and robust quantum computation with bosonic qubits. Published by the American Physical Society 2024
In our model a fixed Hamiltonian acts on the joint Hilbert space of a quantum system and its controller. We show conditions under which measurements, state preparations, and unitary implementations on the system can be performed by quantum operations on the controller only. It turns out that a measurement of the observable AAA and an implementation of the one-parameter group exp(iAr)\mathrm{exp}(\mathrm{iAr})exp(iAr) can be performed by almost the same sequence of control operations. Furthermore, measurement procedures for A+B,A+B,A+B, for (AB+BA),(AB+BA),(AB+BA), and for $i\ensuremath{\lfloor}A,B\ensuremath{\rfloor}$ can be constructed from measurements of AAA and B.B.B. This shows that the algebraic structure of the set of observables can be explained by the Lie group structure of the unitary evolutions on the joint Hilbert space of the measuring device and the measured system. A spin chain model with nearest-neighborhood coupling shows that the border line between controller and system can be shifted consistently.
Fault-tolerant quantum computation with bosonic qubits often necessitates the use of noisy discrete-variable ancillae. In this work, we establish a comprehensive and practical fault-tolerance framework for such a hybrid system and synthesize it with fault-tolerant protocols by combining bosonic quantum error correction (QEC) and advanced quantum control techniques. We introduce essential building blocks of error-corrected gadgets by leveraging ancilla-assisted bosonic operations using a generalized variant of path-independent quantum control. Using these building blocks, we construct a universal set of error-corrected gadgets that tolerate a single-photon loss and an arbitrary ancilla fault for four-legged cat qubits. Notably, our construction requires only dispersive coupling between bosonic modes and ancillae, as well as beam-splitter coupling between bosonic modes, both of which have been experimentally demonstrated with strong strengths and high accuracy. Moreover, each error-corrected bosonic qubit is comprised of only a single bosonic mode and a three-level ancilla, featuring the hardware efficiency of bosonic QEC in the full fault-tolerant setting. We numerically demonstrate the feasibility of our schemes using current experimental parameters in the circuit-QED platform. Finally, we present a hardware-efficient architecture for fault-tolerant quantum computing by concatenating the four-legged cat qubits with an outer qubit code utilizing only beam-splitter couplings. Our estimates suggest that the overall noise threshold can be reached using existing hardware. These developed fault-tolerant schemes extend beyond their applicability to four-legged cat qubits and can be adapted for other rotation-symmetrical codes, offering a promising avenue toward scalable and robust quantum computation with bosonic qubits.
A closed quantum system is defined as completely controllable if an arbitrary unitary transformation can be executed using the available controls. In practice, control fields are a source of unavoidable noise. Can one design control fields such that the effect of noise is negligible on the timescale of the transformation? Complete controllability in practice requires that the effect of noise can be suppressed for an arbitrary transformation. The present study considers a paradigm of control, where the Lie-algebraic structure of the control Hamiltonian is fixed, while the size of the system increases, determined by the dimension of the Hilbert space representation of the algebra. We show that for large quantum systems, generic noise in the controls dominates for a typical class of target transformation; i.e., complete controllability is destroyed by the noise.
Quantum Error Recovery (QER) uses knowledge of the error channel acting on a quantum system to find optimal recovery maps. The scheme restores the uncorrupted state with a fidelity exceeding that achieved by noise parameter independent quantum error correction. We use a generic coherent QER map implemented with a quantum circuit acting on the system together with ancillary qubits to recover quantum information stored in permutation invariant (PI) codes. PI codes admit tunable parameters to suit the noise model and benefit from simple recovery operation circuits with reduced addressability requirements, unlike stabilizer codes. We showcase the method by modeling QER in PI codes after collective and local symmetric correlated amplitude-damping (AD) noise, a non-Pauli noise process for which stabilizer codes often require additional overhead. We also propose a new PI code family called CAD codes with explicit examples on 4 and 9 qubits for global symmetric AD errors. We show that CAD9 (supported on 9 qubits) code beats many existing codes by more than one order of magnitude. For the CAD4 code, which perfectly corrects 1 global symmetric AD error, the compiled recovery circuit consists of 10 system and system-ancilla gates which can be realized from linear geometric phase gates. Our work provides a direct path from optimized recovery maps to experimentally implementable, low-overhead protocols.