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Ambiguous stabilizer codes and quantum noise characterization

S. Omkar, R. Srikanth, Subhashish Banerjee

Published Feb 2, 2015• 0 citations
OpenAlex

Abstract

A quantum error correcting is a subspace $\mathcal{C}$ such that allowed errors acting on any state in $\mathcal{C}$ can be corrected. A quantum for which state recovery is only required up to a logical rotation within $\mathcal{C}$, can be used for detection of errors, but not for quantum error correction. Such a with stabilizer structure, which we call an ambiguous stabilizer code (ASC), can nevertheless be useful for the characterization of quantum dynamics (CQD). The use of ASCs can help lower the size of CQD probe states used, but at the cost of increased number of operations.

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A quantum error correcting is a subspace C\mathcal{C}C such that allowed errors acting on any state in C\mathcal{C}C can be corrected. A quantum for which state recovery is only required up to a logical rotation within C\mathcal{C}C, can be used for detection of errors, but not for quantum error correction. Such a with stabilizer structure, which we call an ambiguous stabilizer code (ASC), can nevertheless be useful for the characterization of quantum dynamics (CQD). The use of ASCs can help lower the size of CQD probe states used, but at the cost of increased number of operations.
A quantum error-correcting code is a subspace C\mathcal{C}C such that allowed errors acting on any state in C\mathcal{C}C can be corrected. A quantum code for which state recovery is only required up to a logical rotation within C\mathcal{C}C can be used for the detection of errors, but not for quantum error correction. Such a code with a stabilizer structure, which we call an ``ambiguous stabilizer code'' (ASC), can nevertheless be useful for the characterization of quantum dynamics (CQD). The use of ASCs can help lower the size of CQD probe states used, but at the cost of an increased number of operations.
Characterizing noisy quantum processes is important to quantum computation and communication (QCC), since quantum systems are generally open. To date, all methods of characterization of quantum dynamics (CQD), typically implemented by quantum process tomography, are off-line, i.e., QCC and CQD are not concurrent, as they require distinct state preparations. Here we introduce a method, ``quantum error correction based characterization of dynamics,'' in which the initial state is any element from the code space of a quantum error correcting code that can protect the state from arbitrary errors acting on the subsytem subjected to unknown dynamics. The statistics of stabilizer measurements, with possible unitary preprocessing operations, are used to characterize the noise, while the observed syndrome can be used to correct the noisy state. Our method requires at most $2({4}^{n}\ensuremath{-}1)$ configurations to characterize arbitrary noise acting on nnn qubits.
Realization of long distance quantum cryptography requires quantum error correction (QEC) of the errors introduced by noise on communication channels. This chapter begins with presenting common sources of errors in QEC. For quantum information processing qubits using photons, trapped ions, superconducting circuits and spins in semiconductors have been considered. Coherent quantum errors arise due to the errors associated with system controls and/or characterization where imprecise operation on the qubit introduces inaccurate Hamiltonian dynamics. The basic principle involved in classical error correction is to increase the number of bits more than is required for a given amount of information. Digitization of quantum errors makes it possible to correct any quantum error using the ability to correct a finite set of errors. The use of stabilizer codes to describe QEC codes is highly useful. It allows for easy synthesis of correction circuits and also shows how logical operations can be performed directly on encoded data.
The characterization of quantum devices is crucial for their practical implementation but can be costly in experimental effort and classical postprocessing. Therefore, it is desirable to measure only the information that is relevant for specific applications and develop protocols that require little additional effort. In this Letter, we focus on the characterization of quantum computers in the context of stabilizer quantum error correction. For arbitrary stabilizer codes, subsystem codes, and data syndrome codes, we prove that the logical error channel induced by Pauli noise can be estimated from syndrome data under minimal conditions. More precisely, for any such code, we show that the estimation is possible as long as the code can correct the noise.
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.
Quantum error correction acts as an entropy pump, transferring noise-induced uncertainty from a protected quantum system into syndrome information stored in an auxiliary memory. Repeated operation requires this memory to be cleared which unavoidably contributes to the energetic cost of error correction. Here, we characterise this contribution for subspace quantum error-correcting codes and identify how it depends on the joint structure of the code, the noise, and the representation of the retained syndrome information. Starting from the Knill--Laflamme conditions, we construct an effective syndrome state whose von Neumann entropy sets a lower bound on the ideal work required to maintain a reusable syndrome register. Projective syndrome readout generally generates additional entropy, and we quantify the resulting gap through measurement inefficiency. We then specialise to stabiliser codes under independent local Pauli noise and analyse two classical levels of syndrome representation. At the level of abstract error labels, degeneracies among single-qubit errors reduce the leading-order entropy of processed recovery labels. At the parity-check level, lower-weight checks reduce the marginal entropy generated by individual measurement outcomes in the low-noise regime. We identify the additional burden associated with retaining and separately erasing these outcomes as a bit-level inefficiency, and illustrate both costs for the five-qubit, Steane, generalised Shor, and rotated surface codes. Our results establish a hierarchy of syndrome-memory energetic costs and identify the code, noise, and measurement structures that control the ideal thermodynamic burden of subspace quantum error correction.