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Gentle tomography and efficient universal data compression Caltech IQI Charlie Bennett Aram Harrow Seth Lloyd Jan 14, 2003 Classical data compression • Asymptotically optimal: n copies of {pi} are compressed to n(H({pi})+dn) bits with en error where dn,en!0 as n!1. • Computationally efficient: running time is polynomial in n. • Universal: algorithms work for any i.i.d. source. Example: Huffman coding Map letter i to a prefix-free codeword of length -log pi. 0 A 1 0 1 B 0 C 1 D Letter (i) Probability (pi) Codeword A B C D 1/2 1/4 1/8 1/8 0 10 110 111 The expected number of bits per symbol is -ipilogpi =H(pi) and the standard deviation is O(pn). Quantum Data Compression • A quantum source r=jqj|yjihyj| can be diagonalized r=ipi|iihi| and treated as a classical source. • The main difference between quantum and classical Huffman coding is that measuring the length of the output will damage the state. • Also, Schumacher compression assumes we know the basis in which r is diagonal. Therefore it is optimal and efficient, but not universal. Universal quantum data compression • JHHH showed that for any R, there exists a space Hn,R of dimension 2nR+o(n) such that if S(r)<R then most of the support of rn lies in Hn,R. (quant-ph/9805017) • HM showed that you don’t need to know the entropy. However, they present no efficient implementation. (quant-ph/0209124) • JP give an efficient, universal algorithm, but we like ours better. (quant-ph/0210196) Goal: Efficient, Universal quantum data compression • Modeled after classical Huffman coding. • Pass 1: Read the data, determine the type class (empirical distribution) and decide on a code. • Pass 2: Encode Step 1: Gentle tomography • Problem: Quantum state tomography damages the states it measures. • Solution: use weak measurements. For example, in NMR the average value of sx can be approximately measured for 1020 spins without causing decoherence. Gentle tomography algorithm • Reduce tomography to estimating d2 observables of the form tr rsk. • For each estimate: – Randomly divide the interval 0,…,n into n1/4 different bins. – Measure only the bin that trrsk falls into. n1/4 random bin boundaries ntrrsk n 0 uncertainty O(n1/2) bin width O(n3/4) Gentle tomography lemma • If a projective measurement {Mj} has high probability of obtaining a particular outcome k, then obtaining k leaves the state nearly unchanged. (Ahlswede, Winter) • Thus, if we are very likely to estimate the state correctly, then this estimation doesn’t cause very much damage. Implementation Example of a circuit to gently estimate h1|r|1i. … |0ilog n … rn +1 +1 +1 Weakly Measure -1 -1 -1 Result • An information-disturbance tradeoff curve: error¢disturbance>n-1/2 (up to logarithmic factors). • (Can we prove this is optimal?) • In particular, we can set both error and damage equal to n-1/4log n. Step 2: Compression • Given an estimate s with |r-s|<e, how do we compress rn? Huffman coding with an imperfect state estimate • Suppose we encode rn according to s=iqi|iihi|. • Codeword i has length -log qi and occurs with probability hi|r|ii. • Thus the expected length is i-log qi hi|r|ii = -tr(rlogs) = S(r) + S(r||s). • Unfortunately, S(r||s) is not a continuous function of s. Dealing with imperfect estimates. • Replace s with (1-d)s + dI/d. • This makes the rate no worse than S(r)+elog 1/ed. Result • A polynomial time algorithm that compresses rn into n(S(r)+O(n-slog2n)) qubits with error O(ns-1/2log n). Other methods • Schumacher coding and the JHHH method both achieve S(r)+O(kn-1/2) qubits/signal and exp(-k2) error. • The HM method has roughly the same ratedisturbance trade-off curve, though their overflow probability is lower. • The JP method achieves error n-e and rate S(r)+n-r when e=1/2+r(1+d2). For example, compressing qubits with constant error gives rate S(r)+O(n-1/10). Future directions • Stationary ergodic sources. Likewise, a visible quantum coding theory exists, but the only algorithm is classical Lempel-Ziv. • Proving converses: – No exponentially vanishing error. – No on-the-fly compression. – Information/disturbance and rate/disturbance trade-offs. A Quantum Clebsch-Gordon transform (Bacon & Chuang) • Hn=©l`n Al Bl. l is a partition of n into d parts, Al is an irreducible representation of SU(d) and Bl is an irrep of Sn. • Wanted: an efficient quantum circuit to map |i1…ini!l|li|ali|bli. • Useful for universal compression, state estimation, hypothesis testing, and entanglement concentration/dilution/distillation.