Coherence protection by random coding

INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS

J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S353–S355

doi:10.1088/1464-4266/7/10/022

Coherence protection by random coding E Brion1 , V M Akulin1 , I Dumer2 , G Harel3,4 and G Kurizki5 1 2 3 4 5

Laboratoire Aim´e Cotton, CNRS II, Bˆatiment 505, 91405 Orsay Cedex, France College of Engineering, University of California, Riverside, CA 92521, USA Spinoza Institute, Utrecht University, Leuvenlaan 4, 3508 TD Utrecht, The Netherlands Department of Computing, University of Bradford, Bradford BD7 1DP, UK Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel

Received 31 March 2005, accepted for publication 9 June 2005 Published 21 September 2005 Online at stacks.iop.org/JOptB/7/S353 Abstract We show that the multidimensional Zeno effect combined with non-holonomic control allows one to efficiently protect quantum systems from decoherence by a method similar to classical random coding. The method is applicable to arbitrary error-inducing Hamiltonians and general quantum systems. The quantum encoding approaches the Hamming upper bound for large dimension increases. Applicability of the method is demonstrated with a seven-qubit toy computer. Keywords: coherent control, complex systems, decoherence, error correction, non-holonomic control, random coding, quantum computation, quantum information, quantum mechanics, Zeno effect

Classical random coding is a method for producing efficient linear codes [n, k, d] where k-bit words are encoded as randomly chosen n-bit sequences [1]. With minimum Hamming distance d between any two code words, such a code meets the Varshamov–Gilbert bound k/n
1− H (d/n) in the limit n → ∞, where H (x) = −x log2 x − (1 − x) log2 (1 − x) is the binary entropy function for any x ∈ [0, 1]. In the present paper we show that the idea of random coding is applicable to quantum codes and allows for efficient production, encoding and decoding of such codes. Strong mixing or entanglement occur in the phase space or Hilbert space of complex classical or quantum systems respectively, and in principle can be used for random coding. In practice, however, the dynamics of classical complex systems is not reversible, and therefore subsequent decoding will not be possible. And yet, the dynamics of quantum multi-dimensional systems can be reversible, provided the underlying physical system is simple. The spin-echo phenomenon is a typical example. High dimensionality in simple quantum systems is instrumental in affording the massive parallel computing capacity of quantum computers [2]. In the present paper we show how to harness such high dimensionality to provide coherence protection of generic applicability. To combine strong mixing with reversibility, we consider a quantum system with a large number of separate energy levels and two simple interactions that can be controlled. In such a system we can use non-holonomic control [3, 4] 1464-4266/05/100353+03$30.00 © 2005 IOP Publishing Ltd

to encode quantum data into strongly mixed states by a unitary transformation Cˆ and subsequently to apply the inverse transformation Cˆ −1 to decode the data6 . Encoding the data into many levels allows us to strongly reduce the rate at which the data are corrupted by arbitrary physical errors, as will be shown. In turn, by applying the Zeno effect [5, 6] we can restore the slightly corrupted data back to their original values with high probability. To encode k qubits of data in a high dimensional Hilbert space of n qubits, we introduce n − k ancilla qubits in addition to the data qubits. This results in an increased number of possible errors, which depends polynomially on n, but the error rate can be decreased at will because the infinitesimal errors are semi-orthogonal to the encoded data to a degree exponential in n − k. The degree of semi-orthogonality reflects the error-correction efficacy of the coding. For coding in minimal dimensions [7], efficacy requires a careful choice of the code, but in high dimensions it is trivially achieved by random coding. In physical terms, random coding translates into strong mixing, or full population of all energy levels, which is attainable by non-holonomic control with a number of interaction switchings depending only polynomially on n. Thus the random coding approach of the present paper complements our earlier non-holonomic/Zeno decoherence suppression scheme [4] that demands exponential effort to find 6

The encoding and decoding steps are assumed to be error free, in common with the practice in classical coding theory.

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E Brion et al

and effect an exact code, and which is therefore restricted to low dimension systems. The state of the k data qubits is a continuous complexvalued vector in the entire 2k -dimensional Hilbert space Hk . When the original system is expanded by n − k ancilla qubits in some state |α, ˜ the extended state vectors form a subspace Hk,α˜ = |α ˜ α| ˜ Hn in the bigger space Hn . The quantum encoding Cˆ is a unitary transformation that maps the subspace Hk,α˜ onto a 2k -dimensional code subspace C = Cˆ Hk,α˜ in Hn . Note that C is spanned by the columns of the rectangular matrix Cˆ |α, ˜ and has zero projections on the 2n − 2k column vectors of the matrices Cˆ |α for states |α ⊥ |α ˜ completing a basis for the ancilla Hilbert space. The regular Zeno effect allows one to preserve a quantum system in a known initial state by periodic projective measurements, provided that the period T is short relative to the relaxation time T2 . Here, however, we only perform measurements on the ancilla part in order to protect the unknown quantum state of the data part, restoring it to the initial state up to the second-order terms in T / T2 . To see the results of the Zeno effect, consider first the state vector |S = |α ˜ |s of the compound system formed by the ancilla in the state |α ˜ and the data part in some state |s. The encoding Cˆ maps the state |S to the code vector |v = Cˆ |α ˜ |s ∈ C , which then undergoes an error-induced, uncontrolled unitary evolution of the form  M M 
 ˆ Eˆ m f m (t) dt, (1) e−i Em f m (t) dt  Iˆ − i Uˆ E = m=1

m=1

where the time ordering of the product is implicit7 . The error sources are assumed to be described by Hamiltonians Eˆ m of interactions with external random fields fm (t) that produce an uncontrolled evolution of the system. We also assume that over the Zeno period T the different fields have small actions,  t+T f m (x) dx|  1, so that only the identity operator Iˆ | Eˆ m t and the first-order terms are important in the Taylor series. Now let ρˆsc = |s s| be the density matrix of the data part ˜ is before the action of errors. Then ρˆ = |S S| = |α ˜ ρˆsc α| the initial density matrix of the entire system. The variation ˜ δ ρˆ |α ˜ of the density matrix of the data part δ ρˆsc = α| ˆ followed by error action as a result of the encoding by C, equation (1), decoding with Cˆ −1 and projection on the ancilla state, is given by the commutator   M   −1 f m (t) dt α| ˜ , ρˆsc . (2) ˜ Cˆ Eˆ m Cˆ |α δ ρˆsc = −i m=1

Therefore ρˆsc satisfies the master equation i

  dρˆsc = hˆ e , ρˆsc dt

(3)

with effective Hamiltonian hˆ e =

M 

˜ . fm α| ˜ Cˆ −1 Eˆ m Cˆ |α

(4)

m=1 7 We assume for simplicity that the uncontrolled evolution is unitary;however, in the Zeno regime our protection scheme applies equally well to non-unitary errors.

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This shows that the Zeno effect can protect the data coherence, that is, ensure that ρˆsc remains constant, provided that we have hˆ e = 0. The requirement that the effective Hamiltonian (4) should vanish for arbitrary field strengths fm can be written in terms of the code vectors and error matrices as

v Eˆ m |v = 0; (5)

 ∀m = 1 . . . M, ∀ |v , v ∈ C ; which shows that it is less restrictive than the quantum errorcorrection conditions of [8]:

v Eˆ i Eˆ j |v = αi j δv v ; (6)

 ∀ |v , v ∈ C ; ∀i, j = 1 . . . M. In fact, instead of satisfying the conditions of equation (5) ˆ as is done in exactly using a carefully selected encoding C, the low dimension scheme [4], in high dimension systems it is sufficient to apply a random encoding Cˆ in order to minimize the effective Hamiltonian (4) and thus suppress the error accumulation in the density matrix. For systems with binary interaction an exponential reduction of the error-accumulation rate can be achieved if the projection rate 1/ T grows faster than n 2 . Moreover, the code rate R = k/n tends to unity as 1 − o(n 2 T ). In particular, one can reduce all effective Hamiltonians ˆ ˜ Cˆ −1 Eˆ m Cˆ |α hˆ m = α| ˜ by applying a random encoding C, which allows one to reduce the rate of decoherence by a factor of the order of the ancilla dimension A = 2n−k . Indeed, for individual qubits, a typical single-particle or binary interaction Eˆ m is given by a sparse matrix that strongly couples each state to only a few other states, with most matrix elements being zero. A generic unitary encoding Cˆ smears out this coupling over all 2n states of the entire system, thus producing a typical matrix element of Cˆ −1 Eˆ m Cˆ that is 2n times smaller than the typical nonzero matrix element of Eˆ m . Since only 2k of these states contribute to the projected interactions hˆ m , the overall reduction is equal to 2n−k . From a more general viewpoint, the effectiveness of the random encoding rests on the fact that the decoherence rate is only polynomial in the number of particles n, whereas the dimension of the entire Hilbert space is exponential in n. The random encoding allows one to store quantum information in strongly entangled states of many-body systems that are only weakly affected by physically realistic perturbations. This random encoding also allows one to specify a relation between the dimension K = 2k of the information system and the maximum number t of possible errors caused by binary interactions. Given D single-particle quantum states, l t n  2 D − 1 different possible error there exist M = l=1 l matrices (here D = 2 for qubits). Since the energy of binary interaction scales as n 2 , each error matrix yields a decoherence rate of W = n 2 / T2 . This rate is reduced A = 2n−k times after the matrix elements are smeared out over the entire system. Thus, given n → ∞ identical particles, one can correct t errors with high fidelity if the total decoherence rate W 2k−n M vanishes. This condition, W 2k−n M → 0, gives the Hamming bound [1] (7) 1 − k/n − log2 M 1/n  0.

Coherence protection by random coding

Hence, to asymptotically meet this classical bound, we can randomly choose an (n, k) code, thus leaving the error patterns of weight t or less uncorrected with a vanishing probability. The question arises now as to how to perform a random encoding in an efficient way. One can achieve this feasibly with the help of non-holonomic control, by applying two different natural Hamiltonians Hˆ 1 and Hˆ 2 that satisfy certain conditions [3]. A generic choice of a sequence of nonholonomic control timings t1 , t2 , t3 , . . . , t M of a reasonably short length M will result in a random encoding transformation ˆ ˆ ˆ ˆ Cˆ = e−it M H2 · · · e−it3 H1 e−it2 H2 e−it1 H1 .

(8)

Indeed, the sequential application of two different Hamiltonians suggested by equation (8) results in a completely generic, random transformation provided that: (i) the number M of the control timings tl is of the order of n = log2 (A × K ); (ii) the operators Hˆ 1 and Hˆ 2 satisfy the conditions of nonholonomic control [3]; and (iii) the time intervals tl are long enough to ensure big acquired actions, | Hˆ 1,2 tl |  h¯ . Therefore, given a physical system, the main problem is to find two controllable interactions that satisfy the conditions needed for non-holonomic control, and allow one to perform the decoding transformation Cˆ −1 at the same level of complexity. We present such a pair Hˆ 1 , Hˆ 2 for the example of the molecule proposed in [9] as a toy quantum computer of seven qubits, for which the applicability of both the low dimensional (exact) scheme [4] and the present high dimension (approximative) approach was verified. The first operator is the nuclear magneto-dipole interaction  Bx µ j σˆ x( j ) (9) Hˆ 1 = j

of the molecule in the ground ro-vibronic state |g with a static external magnetic field Bx oriented along the x-axis. The second operator is the Raman coupling   R R (i) ( j ) Hˆ 2 = δω−1 µ(2) ˆ a σˆ b (10) i, j Ba Bb σ i< j a,b=x,y,z

− → in a field B R oscillating at a high frequency ω detuned by δω = ωtr − ω from the frequency ωtr of transition to an exited ro-vibronic state |e. Here µ j = g| µ j (r ) |g is the average value of the coordinate dependent gyromagnetic ratio µ j (r ) of the j th nucleus in the ground state, and µ(2) i, j = g| µi (r ) |e e| µ j (r ) |g represents the matrix element of the Raman transition. By changing the direction of Bx from positive to negative and reversing the detuning δω, one can

vary the signs of the Hamiltonians, which allows one to easily construct the inverse transformation Cˆ −1 just by reversing the control sequence in equation (8) along with effecting the change of the signs, ˆ ˆ ˆ ˆ Cˆ −1 = eit1 H1 · · · eit M −2 H2 eit M −1 H1 eit M H2 .

(11)

Summarizing the main results of the paper, we consider open quantum systems that consist of two parts—the data part and an ancilla. The latter is periodically reset to a given quantum state while the data part is protected against uncontrolled decoherence with accuracy proportional to the period. Initially disentangled, the two parts are driven to a highly entangled state, which is weakly sensitive to the decoherence, by means of unitary non-holonomic control. The inverse unitary transformation brings the system to a state that differs from the initial state only in the ancilla part. After resetting the ancilla, the system returns to the original state. Implemented in a multidimensional system of dimension N , the control sequence needs only log N switches to strongly reduce the decoherence rate.

Acknowledgments The research of VMA was supported by an EC QUACS grant and the research of ID was supported by NSF grant CCR0097125.

References [1] Gallager R G 1968 Information Theory and Reliable Communication (New York: Wiley) [2] Aharonov D 1998 Preprint quant-ph/9812037 [3] Harel G and Akulin V M 1999 Phys. Rev. Lett. 82 1 Akulin V M, Gershkovich V and Harel G 2001 Phys. Rev. A 64 012308 [4] Brion E, Harel G, Kebaili N, Akulin V M and Dumer I 2004 Europhys. Lett. 66 157 [5] See the pioneering paper by Misra S and Sudarshan E C G 1977 J. Math. Phys. 18 756 and later publications that are closer to the specific problem under consideration: Plenio M B, Vedral V and Knight P L 1997 Phys. Rev. A 55 67 Facchi P, Pascazio S, Scardicchio A and Schulman L S 2002 Phys. Rev. A 65 012108 [6] Kofman A G and Kurizki G 2001 Phys. Rev. Lett. 87 270405 [7] Calderbank A R and Shor P W 1996 Phys. Rev. A 54 1098 [8] Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) p 436 [9] Vandersypen L M K, Steffen M, Breyta G, Yannoni C S, Sherwood M H and Chuang I L 2001 Nature 414 883

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