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Structured Near-Optimal Channel- Adapted Quantum Error Correction

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Structured Near-Optimal Channel- Adapted Quantum Error Correction Structured near-optimal channel- adapted quantum error correction The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Fletcher, Andrew, Peter Shor, and Moe Win. “Structured near- optimal channel-adapted quantum error correction.” Physical Review A 77.1 012320 (2008) [16 pages]. © 2008 The American Physical Society. As Published http://dx.doi.org/10.1103/PhysRevA.77.012320 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/67444 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW A 77, 012320 ͑2008͒ Structured near-optimal channel-adapted quantum error correction Andrew S. Fletcher,1,2,* Peter W. Shor,3,† and Moe Z. Win1,‡ 1Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massuchesetts 02139, USA 2MIT Lincoln Laboratory, 244 Wood Sreet, Lexington, Massachusetts 02420, USA 3Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA ͑Received 28 August 2007; published 17 January 2008͒ We present a class of numerical algorithms which adapt a quantum error correction scheme to a channel model. Given an encoding and a channel model, it was previously shown that the quantum operation that maximizes the average entanglement fidelity may be calculated by a semidefinite program ͑SDP͒, which is a convex optimization. While optimal, this recovery operation is computationally difficult for long codes. Fur- thermore, the optimal recovery operation has no structure beyond the completely positive trace-preserving constraint. We derive methods to generate structured channel-adapted error recovery operations. Specifically, each recovery operation begins with a projective error syndrome measurement. The algorithms to compute the structured recovery operations are more scalable than the SDP and yield recovery operations with an intuitive physical form. Using Lagrange duality, we derive performance bounds to certify near-optimality. DOI: 10.1103/PhysRevA.77.012320 PACS number͑s͒: 03.67.Pp, 02.60.Pn I. INTRODUCTION semble average fidelity and entanglement fidelity, respec- tively. A suboptimal method for minimum fidelity, using an All physical implementations of quantum-information- SDP, was proposed in ͓9͔. An analytical approach to channel- processing systems must incorporate a scheme to mitigate adapted recovery based on the pretty-good measurement and the effects of noise. The most common method for quantum ͓ ͔ ͑ ͒ the average entanglement fidelity was derived in 10 . The error correction QEC is analogous to classical digital error main point of each scheme was to improve error corrective correction schemes. The system of interest is encoded into a procedures by adapting to the physical noise process. subspace of a larger quantum system by means of a quantum As in ͓6͔, we choose to focus our channel-adapted efforts code. After passing through a noisy channel, a syndrome on the recovery operation. While channel adaptation can be measurement projects errors onto orthogonal subspaces from advantageous in both the encoding and the recovery opera- which the original quantum state can be recovered. The first tions, the optimization problem has a significantly nicer form quantum error correcting codes demonstrated that such meth- ͓ ͔ when one of the two is held fixed. The numerical tools we ods could correct arbitrary single-qubit errors 1–3 . These develop can be used for either half of the problem; focusing generic methods enabled a whole range of study in quantum on quantum error recovery ͑QER͒ operations illustrates error correction, particularly as it applies to fault-tolerant nearly all of the important numerical procedures. quantum computing. The optimization approach to quantum error recovery The generic approach has its drawbacks, however. Most demonstrates the utility of channel adaptivity. Such efforts notably, quantum codes impose a severe amount of overhead have shown that quantum error correction designed for ge- to correct for arbitrary errors. As an example, the shortest neric errors can be inefficient in the face of a particular noise block code that corrects an arbitrary qubit error embeds one ͓ ͔ process. Since overhead in physical quantum computing de- qubit into five 4,5 . As scaling to many qubits is one of the vices is challenging, it is advantageous to maximize error principal barriers to building a working quantum computer, recovery efficiency. any efforts to improve the efficiency of error recovery are of Recovery operations generated through convex optimiza- great interest. tion methods suffer two significant drawbacks. First, the di- Several recent efforts have explored an optimization- ͓ ͔ mensions of the optimization problem grow exponentially based approach to quantum error recovery 6–9 . In each with the length of the code, limiting the technique to short case, rather than correcting for arbitrary single-qubit errors, codes. Second, the optimal operation, while physically legiti- the error recovery scheme was adapted to a model for the mate, may be quite difficult to implement. The optimization noise, with the goal of maximizing the fidelity of the opera- ͓ ͔ ͑ ͒ routine is constrained to the set of completely positive, trace- tion. In 6 , a semidefinite program SDP was used to maxi- preserving ͑CPTP͒ operations, but is not restricted to more mize the entanglement fidelity, given a fixed encoding and ͓ ͔ ͓ ͔ easily implemented operations. channel model. In 7 and 8 , encodings and decodings were In this paper, we describe efforts to determine near- iteratively improved using the performance criteria of en- optimal channel-adapted quantum error recovery procedures that overcome the drawbacks of optimal recovery. We im- pose an intuitively satisfying structure on the recovery op- *fl[email protected] eration and seek to optimize performance. While still nu- †[email protected] merical procedures, the result is a class of algorithms that is ‡[email protected] less computationally intensive than the SDP and which 1050-2947/2008/77͑1͒/012320͑16͒ 012320-1 ©2008 The American Physical Society FLETCHER, SHOR, AND WIN PHYSICAL REVIEW A 77, 012320 ͑2008͒ be accomplished via the convex optimization routine of semidefinite programming. For the remainder of the paper, ρin Encoder Channel Recovery R◦E(ρin) we will discuss routines to approach the optimum channel- Uc E R adapted recovery through a more computationally feasible method. In several cases, the routines also yield an intuitive form for the recovery operation. E We will make use of a convenient isomorphism in which bounded linear operators are represented by vectors and de- ͉͗ ͘͘ FIG. 1. Quantum error correction block diagram. For channel- noted with the symbol · . While there are several choices ͓ ͔ adapted recovery, the encoding isometry UC and the channel EЈ are for this isomorphism 11,12 , including most intuitively a considered as a fixed operation E and the recovery R is chosen “stacking” operation, we will follow the conventions of ͓13͔ according to the design criteria. ͑also ͓9͔͒, which result in an isomorphism that is indepen- dent of the choice of basis. For convenience, we will restate yields recovery operations of an intuitive and potentially re- the relevant results here. ͚ ͉ ͗͘ ͉ H alizable form. Let A= ijaij i j be a bounded linear operator from to K ͑i.e., AʦL͑K,H͒͒, where ͕͉i͖͘ and ͕͉j͖͘ are bases for K be the dual of H. This is also a ءII. CHANNEL-ADAPTED RECOVERY and H, respectively. Let H Hilbert space, generally understood as the space of bras ͗j͉. To adapt quantum error recovery to a specific channel If we relabel the elements as ͉j͘=͗j͉, then we represent A as ء .model, we must first determine a measure of performance ͓ ͔ ͑ ͒ a vector in the space K H as As detailed in 6,7 , both entanglement fidelity Fe and en- ¯ semble average fidelity ͑F͒ yield convex optimization ͉͗ ͘͘ ͉ ͉͘ ͘ ͑ ͒ 1 A = ͚ aij i j . 2 problems. As both measures may be of interest, we will use ij the average entanglement fidelity of ͓10͔, of which either of the above is a special case. Average entanglement fidelity is It is useful to note the following facts. The inner product A ͕␳ ͖ ͗͗ ͉ ͘͘ † defined for a channel and an ensemble E of states i with A B is the Hilbert-Schmidt inner product trA B. Also, the ͉͗ ͗͗͘͘ ͉͘ † prior probabilities pi as partial trace trK A B =AB . Finally, index manipulation yields the relation A B¯͉͗C͘͘=͉͗ACB†͘͘, where B¯ is the con- ¯F ͑E A͒ ͚ p F ͑␳ A͒ ͚ p ͉ ͑␳ A ͉͒2 ͑ ͒ e , = i e i, = i tr i k , 1 ¯ i i,k jugate of B such that B͉␺͘=B͉␺͘ for all ͉␺͘. These relations lead directly to a convenient representa- ͕ ͖ A L͑K͒ in terms of aۋwhere Ak are the Kraus elements for the CPTP map . tion of a CPTP operation A:L͑H͒ ء Consider the simple block diagram for a QEC system positive semidefinite ͑PSD͒ operator XA ʦL͑K H ͒ EЈ given in Fig. 1. We begin with a fixed model, labeled ,to ͓11,12,14–16͔. The PSD operator is calculated from the describe the physical noise process. In designing a QEC pro- ͕ ͖ A Kraus elements Ak of as cedure, we can choose the encoding UC and the recovery R operation . By holding either the encoding or the recovery ͉͗ ͗͗͘͘ ͉͘ ͑ ͒ XA = ͚ Ak Ak . 3 operation as fixed, optimizing the other can be cast as a k convex optimization problem ͓6–8͔. As done in ͓7,8͔, one can iteratively optimize a encoding and recovery scheme.
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