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arXiv:quant-ph/0509075 · Physical Review A

Feed-forward and its role in conditional linear optical quantum dynamics

Stefan Scheel, William J. Munro, Jens Eisert, Kae Nemoto, Pieter Kok

Published Mar 7, 2006• 14 citations
arXivHTMLPDF

Abstract

Nonlinear optical quantum gates can be created probabilistically using only single-photon sources, linear optical elements, and photon-number-resolving detectors. These gates are heralded but operate with probabilities much less than 1. There is currently a large gap between the performance of the known circuits and the established upper bounds on their success probabilities. One possibility for increasing the probability of success of such gates is feed-forward, where one attempts to correct certain failure events that occurred in the gate's operation. In this Brief Report we examine the role of feed-forward in improving the success probability. In particular, for the nonlinear sign-shift gate, we find that in a three-mode implementation with a single round of feed-forward the optimal average probability of success is approximately given by ${p}_{\mathrm{success}}=0.272$. This value is only slightly larger than the general optimal success probability without feed-forward, ${p}_{\mathrm{success}}=0.25$.

DOI
10.1103/physreva.73.034301

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Nonlinear optical quantum gates can be created probabilistically using only single-photon sources, linear optical elements, and photon-number-resolving detectors. These gates are heralded but operate with probabilities much less than 1. There is currently a large gap between the performance of the known circuits and the established upper bounds on their success probabilities. One possibility for increasing the probability of success of such gates is feed-forward, where one attempts to correct certain failure events that occurred in the gate's operation. In this Brief Report we examine the role of feed-forward in improving the success probability. In particular, for the nonlinear sign-shift gate, we find that in a three-mode implementation with a single round of feed-forward the optimal average probability of success is approximately given by psuccess=0.272{p}_{\mathrm{success}}=0.272psuccess​=0.272. This value is only slightly larger than the general optimal success probability without feed-forward, psuccess=0.25{p}_{\mathrm{success}}=0.25psuccess​=0.25.
We have successfully used fast electronic feed-forward to increase the probability of success of a linear optical implementation of a programmable phase gate from 25% to its theoretical limit of 50%. The feed-forward applies a conditional unitary operation, which changes the incorrect output states of the data qubit to the correct ones. The gate itself rotates an arbitrary quantum state of the data qubit around the zzz axis of the Bloch sphere with the angle of rotation being fully determined by the state of the program qubit. The gate implementation is based on fiber optics components. Qubits are encoded into spatial modes of single photons. The signal from the feed-forward detector is led directly to a phase modulator using only a passive voltage divider. We have verified the increase of the probability of success and characterized the gate operation by means of quantum process tomography. We have demonstrated that the use of feed-forward affects neither the process fidelity nor the output-state fidelities.
The fundamental gates of linear optics quantum computation are realized by using single photons sources, linear optics, and photon counters. Success of these gates is conditioned on the pattern of photons detected without using feedback. Here it is shown that the maximum probability of success of these gates is typically strictly less than 1. For the one-mode nonlinear sign shift, the probability of success is bounded by 1/2. For the conditional sign shift of two modes, this probability is bounded by 3/4. These bounds are still substantially larger than the highest probabilities shown to be achievable so far, which are 1/4 and 2/27, respectively.
One of the main requirements in linear optics quantum computing is the ability to perform single-qubit operations that are controlled by classical information fed forward from the output of single-photon detectors. These operations correspond to predetermined combinations of phase corrections and bit flips that are applied to the postselected output modes of nondeterministic quantum logic devices. Corrections of this kind are required in order to obtain the correct logical output for certain detection events, and their use can increase the overall success probability of the devices. In this paper, we report on the experimental demonstration of the use of this type of feed-forward system to increase the probability of success of a simple nondeterministic quantum logic operation from approximately 14\frac{1}{4}41​ to 12.\frac{1}{2}.21​. This logic operation involves the use of one target qubit and one ancilla qubit which, in this experiment, are derived from a parametric down-conversion photon pair. Classical information describing the detection of the ancilla photon is fed forward in real time and used to alter the quantum state of the output photon. A fiber-optic delay line is used to store the output photon until a polarization-dependent phase shift can be applied using a high-speed Pockels cell.
In this Letter, the problem of finding optimal success probabilities of linear optics quantum gates is linked to the theory of convex optimization. It is shown that by exploiting this link, upper bounds for the success probability of networks realizing single-mode gates can be derived, which hold in generality for postselected networks of arbitrary size, any number of auxiliary modes, and arbitrary photon numbers. As a corollary, the previously formulated conjecture is proven that the optimal success probability of a nonlinear sign shift without feedforward is 1/4, a gate playing the central role in the scheme of Knill-Laflamme-Milburn for quantum computation. The concept of Lagrange duality is shown to be applicable to provide rigorous proofs for such bounds, although the original problem is a difficult nonconvex problem in infinitely many objective variables. The versatility of this approach is demonstrated.
By using the abstract linear-optical network derived by Scheel and Lütkenhaus (2004 New J. Phys. 6 51) we show that for the lowest possible ancilla photon numbers the probability of success of realizing a (single-shot) generalized nonlinear sign-shift gate on an ( N + 1)-dimensional signal state scales as 1/ N 2 . We limit ourselves to single-shot gates without conditional feed-forward. We derive our results by using determinants of Vandermonde-type over a polynomial basis which is closely related to the well-known Jacobi polynomials.
Two qubit gates for photons are generally thought to require exotic materials with huge optical nonlinearities. We show here that, if we accept two qubit gates that only work conditionally, single photon sources, passive linear optics and particle detectors are sufficient for implementing reliable quantum algorithms. The conditional nature of the gates requires feed-forward from the detectors to the optical elements. Without feed forward, non-deterministic quantum computation is possible. We discuss one proposed single photon source based on the surface acoustic wave guiding of single electrons.