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Scaling of success probabilities for linear optics gates

Scheel, S

Published Oct 1, 2004• 9 citations
OpenAlex

Abstract

By using the abstract linear-optical network derived in [S. Scheel and N. L\\"utkenhaus, New J. Phys. \\textbf{6}, 51 (2004)] we show that for the lowest possible ancilla photon numbers the probability of success of realizing a generalized nonlinear sign shift gate on an ($N+1$)-dimensional signal state does not exceed $1/N^2$. This result is underlined by a scaling argument that shows that for any choice of photon numbers the probability scales as $1/N^3$ on average. The analytical derivation involves mathematical methods such as generalized Vandermonde determinants over certain polynomial bases and symmetric polynomials.

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Scaling of success probabilities for linear optics gates

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By using the abstract linear-optical network derived in [S. Scheel and N. L\"utkenhaus, New J. Phys. \textbf{6}, 51 (2004)] we show that for the lowest possible ancilla photon numbers the probability of success of realizing a generalized nonlinear sign shift gate on an (N+1N+1N+1)-dimensional signal state does not exceed 1/N21/N^21/N2. This result is underlined by a scaling argument that shows that for any choice of photon numbers the probability scales as 1/N31/N^31/N3 on average. The analytical derivation involves mathematical methods such as generalized Vandermonde determinants over certain polynomial bases and symmetric polynomials.
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.
We develop an abstract way of defining linear-optics networks designed to perform quantum information tasks such as quantum gates. We will be mainly concerned with the non-linear sign shift (NSS) gate, but it will become obvious that all other gates can be treated in a similar manner. The abstract scheme is extremely well suited for analytical as well as numerical investigations since it reduces the number of parameters for a general setting. With that, we show numerically and partially analytically for a wide class of states that the success probability of generating a NSS gate does not exceed 1/4, which to our knowledge is the strongest bound to date.
We shortly review the basic ideas of generating nonlinear operators with single‐photon sources, linear optical elements, and appropriate measurements of auxiliary modes. This theory allows for an algorithmic construction of useful single‐mode and multi‐mode quantum gates necessary for all‐optical quantum information processing. From this amplitude‐operator point of view all known schemes that realize the nonlinear sign shift gate can be generated, and generalizations to some multi‐mode gates are easily obtained.We will use these ideas of measurement‐induced nonlinearities to develop an abstract way of defining linear‐optics networks. That allows us to treat the problem of maximizing the success probability of generating these conditional quantum gates in a semi‐analytical way. In particular, we show that for a wide class of auxiliary states the success probability of generating a nonlinear sign shift gate does not exceed 1/4 which is the strongest bound to date. Particularly interesting is the observation that the Hilbert space dimension of the ancilla state can be kept to a minimum. We generalize the theory to show that the nonlinear sign shift on single‐mode states with up to N photons can be performed with a success probability that scales as 1/N2.
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.
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.
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.