In a recent paper ([1]=quant-ph/0606035) it is shown how the optimal recovery
operation in an error correction scheme can be considered as a semidefinite
program. As a possible future improvement it is noted that still better error
correction might be obtained by optimizing the encoding as well. In this note
we present the result of such an improvement, specifically for the four-bit
correction of an amplitude damping channel considered in [1]. We get a strict
improvement for almost all values of the damping parameter. The method (and the
computer code) is taken from our earlier study of such correction schemes
(quant-ph/0307138).Comment: 2 pages, 1 figur

Audenaert, K.

Reimpell, M.

Werner, R. F.

English

arXiv

In a recent paper ([1]=quant-ph/0606035) it is shown how the optimal recovery operation in an error correction scheme can be considered as a semidefinite program. As a possible future improvement it is noted that still better error correction might be obtained by optimizing the encoding as well. In this note we present the result of such an improvement, specifically for the four-bit correction of an amplitude damping channel considered in [1]. We get a strict improvement for almost all values of the damping parameter. The method (and the computer code) is taken from our earlier study of such correction schemes(quant-ph/0307138)

Royal Holloway - Pure

arXiv:quant-ph/0606059v1  7 Jun 2006Comment on “Optimum Quantum Error Recovery using Semidefinite Programming”M. Reimpell,1 R. F. Werner,1 and K. Audenaert21Inst. Math. Phys., TU-Braunschweig, Mendelssohnstraße 3, D-38106 Braunschweig, Germany2Department of Physics, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, UKIn a recent paper ([1]=quant-ph/0606035) it is shown how the optimal recovery operation inan error correction scheme can be considered as a semidefinite program. As a possible futureimprovement it is noted that still better error correction might be obtained by optimizing theencoding as well. In this note we present the result of such an improvement, specifically for thefour-bit correction of an amplitude damping channel considered in [1]. We get a strict improvementfor almost all values of the damping parameter. The method (and the computer code) is taken fromour earlier study of such correction schemes (quant-ph/0307138).I. INTRODUCTIONIn a recent paper Fletcher, Shor and Win [1] analyzeerror correction schemes obtained by ab initio optimiza-tion, rather than the adaptation of classical coding tech-niques. The basic idea [2, 3, 4, 5, 6] is that both en-coding E and recovery (or decoding) R can be arbitrarychannels, and that for a given number n of invocationsof a noisy discrete memoryless channel the objective isto bring the channel EN⊗nR as close to the identity aspossible. If the so-called channel fidelity used as a fig-ure of merit (as in [1] and [2]) the optimization is clearlya semidefinite problem, by virtue of Jamiolkowski-Choiduality.In [2] we presented an alternative algorithm for thissemidefinite problem, and used it to generate optimalcodes for various noisy channels, by alternatingly opti-mizing the encoding channel and the recovery channel. In[1] only the recovery is optimized, which already gives amarked improvement some over previously known codes,specifically for the case of the four-bit correction of theamplitude damping channel [8]. The possibility of furtherimprovements by optimizing also the encoding is notedin the discussion (citing also [2]).It so turns out that the required computation (evenfor the same test case) was already done in the autumnof 2003 in a collaboration between the first two authorsand the third author of this note, with the aim of check-ing the power-iteration method of [2] against the betterestablished semidefinite method. Since these results di-rectly support the perspective forwarded in [1], we felt itappropriate to make them immediately available.II. RESULTSFor the theoretical background we refer to either [2] or[1]. In both papers similar ideas and notations are used,so they should be readily accessible from each other. Theamplitude damping channel is the qubit channel N = Nγwith Kraus operatorsK0 =(1 00√1 − γ)and K1 =(0√γ0 0). (1)0 0,2 0,4 0,6 0,8 1damping probability0,20,40,60,81channel fidelityabcFIG. 1: Error correction results for the amplitude dampingchannel (four copies) using three methods: (a) no coding, (b)optimized decoding by Fletcher et al. [1] (with encoding byLeung et al. [8]) and (c) our iterative optimization of both,encoding and decoding [2].These channels form a semigroup (NγNε = Nγε), whichcontracts to the first “unexcited” basis state. The chan-nel fidelity of a noisy channel N is defined asF (N ) := 〈Ω|(N ⊗ id)(|Ω〉〈Ω|)|Ω〉, (2)where Ω is a maximally entangled vector. Note that thisrequires input and output of the channel to be systemswith the same Hilbert space, which is adapted to com-paring the channel with the identity, which is the uniquechannel with F (N ) = 1. This is also closely related [10]to the average fidelity for pure input states (with theaverage taken according to the unitarily invariant mea-sure). The main virtue of choosing this fidelity as a figurecharacterizing the deviation from the identity is that itis linear in N . Such a linear criterion is possible onlybecause the ideal channel is on the boundary of the setof channels.The curve γ 7→ F (Nγ) is the dotted line in Fig. 1.The other lines represent γ 7→ F (EγN⊗4γ Rγ) for variouschoices of Eγ and Rγ . The dashed line uses the encoding2Eγ by Leung et al. [8]|0〉L =1√2(|0000〉+ |1111〉),|1〉L =1√2(|0011〉+ |1100〉),with an optimized decoding Rγ [1]. In fact, we computedthis line by our routines, and it coincides to within pixelresolution with the graph in [1]. The solid line is theresult of the iteration [2], in which Eγ and Rγ are al-ternatingly optimized, keeping the other operation fixed.This iteration has a strict improvement over the Leungcode for 0 < γ < 1.III. DISCUSSIONThe methods in [1] and [2] have the following charac-teristic features:• For a known channel these methods yield excellentresults, without using any special properties of thechannel like symmetry etc..• The optimization of either encoding or decoding isa semidefinite problem for which the solution is acertified global optimum. The process of alternat-ingly optimizing these therefore improves the ob-jective in every step, and hence converges to a lo-cal optimum. However, there is no guarantee forhaving found the global optimum.• The methods suffer from the familiar explosion ofdifficulty as the system size is increased. Correc-tion schemes like the five qubit code can still behandled on a PC, but a nine qubit code would in-volve optimization over 210 × 210-matrices, whichis set up by multiplying and contracting some ma-trices in 218 dimensions. This may be possible onlarge machines, but it is clear that these methodsare useless for asymptotic questions.• The iteration method replacing the semidefinitepackage in [2] has a slight advantage here, becauseit works with a fixed number of Kraus operators.So for the encoding one can put in by hand an iso-metric encoding, which, as our study shows, is oftenoptimal. This cuts down on dimensions, at least forthe optimization of encodings.• For asymptotic coding theory one still needs codes,which can be described also for very large di-mensions, be it by explicit parameterization or bya characterization of typical instances of randomcodes. It is here that methods transferred fromclassical coding theory will continue to be useful.[1] A. S. Fletcher, P. W. Shor and M. Z. Win, “Opti-mum Error Recovery using Semidefinite Programming”,quant-ph/0606035 (2006)[2] M. Reimpell and R. F. Werner, Phys. Rev. Lett. 94,080501 (2005) (also quant-ph/0307138).[3] M. Reimpell, “Quantum error correction as a semi-definite programme”, 26th A2 meeting, Potsdam (2003)[4] M. Reimpell and R. F. Werner, “Quantum Error Cor-recting Codes - Ab Initio”, DPG Spring Conference,München (2004)[5] M. Reimpell, “Ab initio optimization of quantum codes”,PhD Thesis, in preparation[6] M. Reimell and R. F. Werner, “Iterative Optimizationof Quantum Error Correcting Codes”, QIS Workshop,Cambridge (2004)[7] K. Audenaert and B. De Moor, Phys. Rev. A 65030302(R) (2002)[8] D. W. Leung, M. A. Nielsen, I. L. Chuang and Y. Ya-mamoto, Phys. Rev. A 56, (1997)[9] B. Schumacher, Phys. Rev. A 54, 2614, (1996)[10] M. Horodecki, P. Horodecki and R. Horodecki,Phys. Rev. A 60, 1888, (1999)

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