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arXiv:2208.04427 · Quantum

Recovery With Incomplete Knowledge: Fundamental Bounds on Real-Time Quantum Memories

Arshag Danageozian

Published Dec 4, 2023• 5 citations
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Abstract

The recovery of fragile quantum states from decoherence is the basis of building a quantum memory, with applications ranging from quantum communications to quantum computing. Many recovery techniques, such as quantum error correction, rely on the apriori knowledge of the environment noise parameters to achieve their best performance. However, such parameters are likely to drift in time in the context of implementing long-time quantum memories. This necessitates using a "spectator" system, which estimates the noise parameter in real-time, then feed-forwards the outcome to the recovery protocol as a classical side-information. The memory qubits and the spectator system hence comprise the building blocks for a real-time (i.e. drift-adapting) quantum memory. In this article, I consider spectator-based (incomplete knowledge) recovery protocols as a real-time parameter estimation problem (generally with nuisance parameters present), followed by the application of the "best-guess" recovery map to the memory qubits, as informed by the estimation outcome. I present information-theoretic and metrological bounds on the performance of this protocol, quantified by the diamond distance between the "best-guess" recovery and optimal recovery outcomes, thereby identifying the cost of adaptation in real-time quantum memories. Finally, I provide fundamental bounds for multi-cycle recovery in the form of recurrence inequalities. The latter suggests that incomplete knowledge of the noise could be an advantage, as errors from various cycles can cohere. These results are illustrated for the approximate [4,1] code of the amplitude-damping channel and relations to various fields are discussed.

DOI
10.22331/q-2023-12-04-1195

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The recovery of fragile quantum states from decoherence is the basis of building a quantum memory, with applications ranging from quantum communications to quantum computing. Many recovery techniques, such as quantum error correction, rely on the apriori knowledge of the environment noise parameters to achieve their best performance. However, such parameters are likely to drift in time in the context of implementing long-time quantum memories. This necessitates using a "spectator" system, which estimates the noise parameter in real-time, then feed-forwards the outcome to the recovery protocol as a classical side-information. The memory qubits and the spectator system hence comprise the building blocks for a real-time (i.e. drift-adapting) quantum memory. In this article, I consider spectator-based (incomplete knowledge) recovery protocols as a real-time parameter estimation problem (generally with nuisance parameters present), followed by the application of the "best-guess" recovery map to the memory qubits, as informed by the estimation outcome. I present information-theoretic and metrological bounds on the performance of this protocol, quantified by the diamond distance between the "best-guess" recovery and optimal recovery outcomes, thereby identifying the cost of adaptation in real-time quantum memories. Finally, I provide fundamental bounds for multi-cycle recovery in the form of recurrence inequalities. The latter suggests that incomplete knowledge of the noise could be an advantage, as errors from various cycles can cohere. These results are illustrated for the approximate [4,1] code of the amplitude-damping channel and relations to various fields are discussed.
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Quantum process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be Θ(din3dout3/ε2)Θ(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)Θ(din3​dout3​/ε2), where dind_{\mathrm{in}}din​ and doutd_{\mathrm{out}}dout​ are the channel input and output dimensions, respectively, and ε\varepsilonε is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires Ω(din3dout3/ε2)Ω(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)Ω(din3​dout3​/ε2) queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound O(din3dout3/ε2)O(d_{\mathrm{in}}^3 d_{\mathrm{out}}^3/\varepsilon^2)O(din3​dout3​/ε2). Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case din=1d_{\mathrm{in}}=1din​=1. By contrast, coherent protocols with quantum memory achieve query complexity Θ(din2dout2/ε2)Θ(d_{\mathrm{in}}^2 d_{\mathrm{out}}^2/\varepsilon^2)Θ(din2​dout2​/ε2). Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.
Abstract: Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct errors and a formal proof that the classical bounds on the probability of errorcorrecting codes applies to error correcting quantum codes, provided that the interaction is dominated by an identity component.
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.
Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.