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arXiv:1204.6572 · Physical Review A

Quantum stabilizer codes embedding qubits into qudits

Carlo Cafaro, Federico Maiolini, Stefano Mancini

Published Aug 8, 2012• 40 citations
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Abstract

We study, by means of the stabilizer formalism, a quantum error correcting code which is alternative to the standard block codes since it embeds a qubit into a qudit. The code exploits the noncommutative geometry of discrete phase space to protect the qubit against both amplitude and phase errors. The performance of this code is evaluated on Weyl channels by means of the entanglement fidelity as a function of the error probability. A comparison with standard block codes, like five- and seven-qubit stabilizer codes, shows its superiority.

DOI
10.1103/physreva.86.022308

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We study, by means of the stabilizer formalism, a quantum error correcting code which is alternative to the standard block codes since it embeds a qubit into a qudit. The code exploits the noncommutative geometry of discrete phase space to protect the qubit against both amplitude and phase errors. The performance of this code is evaluated on Weyl channels by means of the entanglement fidelity as a function of the error probability. A comparison with standard block codes, like five- and seven-qubit stabilizer codes, shows its superiority.
We study the performance of common quantum stabilizer codes in the presence of asymmetric and correlated errors. Specifically, we consider the depolarizing noisy quantum memory channel and perform quantum error correction via the five- and seven-qubit stabilizer codes. We characterize these codes by means of the entanglement fidelity as a function of the error probability and the degree of memory. We show that their performances are lowered by the presence of correlations, and we compute the error probability threshold values for code effectiveness. Furthermore, we uncover that the asymmetry in the error probabilities does not affect the performance of the five-qubit code, while it does affect the performance of the seven-qubit code, which results in being less effective when considering correlated and symmetric depolarizing errors but more effective for correlated and asymmetric errors.
Using the stabilizer formalism we construct the minimal code into a DDD-dimensional Hilbert space (qudit) to protect a qubit against phase damping. The effectiveness of this code is then studied by means of input-output fidelity.
The scheme of entanglement-assisted quantum error-correcting (EAQEC) codes assumes that the ebits of the receiver are error free. In practical situations, errors on these ebits are unavoidable, which diminishes the error-correcting ability of these codes. We consider two different versions of this problem. We first show that any (nondegenerate) standard stabilizer code can be transformed into an EAQEC code that can correct errors on the qubits of both sender and receiver. These EAQEC codes are equivalent to standard stabilizer codes, and hence the decoding techniques of standard stabilizer codes can be applied. Several EAQEC codes of this type are found to be optimal. In a second scheme, the receiver uses a standard stabilizer code to protect the ebits, which we call a ``combination code.'' The performances of different quantum codes are compared in terms of the channel fidelity over the depolarizing channel. We give a formula for the channel fidelity over the depolarizing channel (or any Pauli error channel), and show that it can be efficiently approximated by a Monte Carlo calculation. Finally, we discuss the tradeoff between performing extra entanglement distillation and applying an EAQEC code with imperfect ebits.
Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.
We develop the theory of entanglement-assisted quantum error-correcting (EAQEC) codes, a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to preshared entanglement. Conventional stabilizer codes are equivalent to self-orthogonal symplectic codes. In contrast, EAQEC codes do not require self-orthogonality, which greatly simplifies their construction. We show how any classical binary or quaternary block code can be made into an EAQEC code. We provide a table of best known EAQEC codes with code length up to 10. With the self-orthogonality constraint removed, we see that the distance of an EAQEC code can be better than any standard quantum error-correcting code with the same fixed net yield. In a quantum computation setting, EAQEC codes give rise to catalytic quantum codes, which assume a subset of the qubits are noiseless. We also give an alternative construction of EAQEC codes by making classical entanglement-assisted codes coherent.
We construct stabilizer states and error-correcting codes on hybrid discrete- and continuous-variable systems, generalizing the Gottesman-Kitaev-Preskill (GKP) lattice formalism. Our framework embeds qudit phase space into a hybrid phase space parameterized by continuous variables, with unit cells that grow with qudit dimension—enabling simultaneous measurement of an arbitrarily large range of noncommuting displacements. Simple hybrid states arise from applying a conditional displacement to a GKP state and a Pauli eigenstate, or by encoding qudits of a stabilizer state into a GKP code; their oscillator-qudit entanglement is non-Gaussian, distinguishing them as a resource beyond tensor products of stabilizer states. Simple hybrid codes are subsystem GKP codes whose gauge factor is entangled with a qudit. Numerical experiments suggest that they can outperform GKP codes against physical noise, with decoders tunable to either qudit or oscillator errors. We also relate stabilizer codes to noncommutative tori, showing that a general construction yields multimode multiqudit GKP extensions. We compute the codes’ logical dimension and operators via Morita equivalence.