A Simple Comparative Analysis of Exact and Approximate Quantum Error Correction

Cafaro, Carlo; Loock, Peter van

doi:10.1142/s1230161214500024

Cited by 7 publications

(7 citation statements)

References 25 publications

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“…As a consequence, the search algorithm can be adapted to the target in order to improve the efficiency of the searching scheme. We emphasize that these considerations are reminiscent of what happens in channel-adapted quantum error correction [36][37][38] and adaptive quantum computing [39] where classical learning techniques can be used to enhance the performance of certain quantum tasks [40,41]. For the time-being, returning to our discussion, we assume that the probability density function ρ (non-uniform) w (θ 1 ,..., θ 2N −1 ) will be denoted as ρ w (θ) and will depend only the coordinate θ 1 def = θ, while it is uniform with respect to the remaining coordinates θ 2 ,..., θ 2N −1 .…”

Section: B Limitations and Possible Future Developmentsmentioning

confidence: 98%

“…(38) In this Appendix we derive Eq. (38). Specifically, exploiting standard trigonometric relations in a clever sequential order, we obtain P (α) = Ã…”

Section: B Limitations and Possible Future Developmentsmentioning

confidence: 99%

“…( 36), ( 33), (22), and (23), the transition probability P ( α) in Eq. (38) becomes (39) From Eq. ( 39), it follows that the maximum P max = P (t * ) of P (t) occurs at the instant t * ,…”

Section: Transition Probabilitymentioning

confidence: 99%

“…(38) In this Appendix we derive Eq. (38). Specifically, exploiting standard trigonometric relations in a clever sequential order, we obtain In this Appendix, we discuss the choice of numerical values of the quantum overlap x by considering non-uniform probability densities of target states on the Bloch sphere [34].…”

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confidence: 99%

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Transition probabilities in generalized quantum search Hamiltonian evolutions

Gassner

Cafaro

Capozziello

2019

Int. J. Geom. Methods Mod. Phys.

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A relevant problem in quantum computing concerns how fast a source state can be driven into a target state according to Schrödinger's quantum mechanical evolution specified by a suitable driving Hamiltonian. In this paper, we study in detail the computational aspects necessary to calculate the transition probability from a source state to a target state in a continuous time quantum search problem defined by a multi-parameter generalized time-independent Hamiltonian. In particular, quantifying the performance of a quantum search in terms of speed (minimum search time) and fidelity (maximum success probability), we consider a variety of special cases that emerge from the generalized Hamiltonian. In the context of optimal quantum search, we find it is possible to outperform, in terms of minimum search time, the well-known Farhi-Gutmann analog quantum search algorithm. In the context of nearly optimal quantum search, instead, we show it is possible to identify sub-optimal search algorithms capable of outperforming optimal search algorithms if only a sufficiently high success probability is sought. Finally, we briefly discuss the relevance of a tradeoff between speed and fidelity with emphasis on issues of both theoretical and practical importance to quantum information processing.

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Section: B Limitations and Possible Future Developmentsmentioning

confidence: 98%

“…(38) In this Appendix we derive Eq. (38). Specifically, exploiting standard trigonometric relations in a clever sequential order, we obtain P (α) = Ã…”

Section: B Limitations and Possible Future Developmentsmentioning

confidence: 99%

“…( 36), ( 33), (22), and (23), the transition probability P ( α) in Eq. (38) becomes (39) From Eq. ( 39), it follows that the maximum P max = P (t * ) of P (t) occurs at the instant t * ,…”

Section: Transition Probabilitymentioning

confidence: 99%

mentioning

confidence: 99%

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Transition probabilities in generalized quantum search Hamiltonian evolutions

Gassner

Cafaro

Capozziello

2019

Int. J. Geom. Methods Mod. Phys.

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“…This code was shown to be approximately correctable for amplitude damping noise, both in terms of worst case fidelity [23] as well as entanglement fidelity [30]. The code is approximate in the sense it does not satisfy the conditions for perfect quantum error correction [19], for any single-qubit error.…”

Section: State Transfer Protocol Based On Adaptive Qecmentioning

confidence: 99%

Pretty good state transfer via adaptive quantum error correction

Jayashankar

Mandayam

2018

Phys. Rev. A

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We examine the role of quantum error correction (QEC) in achieving pretty good quantum state transfer over a class of 1-d spin Hamiltonians. Recasting the problem of state transfer as one of information transmission over an underlying quantum channel, we identify an adaptive QEC protocol that achieves pretty good state transfer. Using an adaptive recovery and approximate QEC code, we obtain explicit analytical and numerical results for the fidelity of transfer over ideal and disordered 1-d Heisenberg chains. In the case of a disordered chain, we study the distribution of the transition amplitude, which in turn quantifies the stochastic noise in the underlying quantum channel. Our analysis helps us to suitably modify the QEC protocol so as to ensure pretty good state transfer for small disorder strengths and indicates a threshold beyond which QEC does not help in improving the fidelity of state transfer.

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Theoretical analysis of a nearly optimal analog quantum search

Cafaro

Alsing

2019

Phys. Scr.

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We analyze the possibility of modifying the original Farhi-Gutmann Hamiltonian algorithm in order to speed up the procedure for producing a suitably distributed unknown normalized quantum mechanical state. Such a modification is feasible provided only a nearly optimal fidelity is sought.We propose to select the lower bounds of the nearly optimal fidelity values such that their deviations from unit fidelity are less than the minimum error probability characterizing the optimum ambiguous discrimination scheme between the two nonorthogonal quantum states yielding the chosen nearly optimal fidelity values. Departing from the working assumptions of perfect state overlap and uniform distribution of the target state on the unit sphere in N -dimensional complex Hilbert space, we determine that the modified algorithm can indeed outperform the original analog counterpart of a quantum search algorithm. This performance enhancement occurs in terms of speed for a convenient choice of both the ratio γ = E /E between the energy eigenvalues E and E of the modified search Hamiltonian and the quantum mechanical overlap x between the source and the target states. Finally, we briefly discuss possible analytical improvements of our investigation together with its potential relevance in practical quantum engineering applications.

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A Simple Comparative Analysis of Exact and Approximate Quantum Error Correction

Cited by 7 publications

References 25 publications

Transition probabilities in generalized quantum search Hamiltonian evolutions

Transition probabilities in generalized quantum search Hamiltonian evolutions

Pretty good state transfer via adaptive quantum error correction

Theoretical analysis of a nearly optimal analog quantum search

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