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arXiv:quant-ph/0109074 · Physical Review A

Generic quantum block compression

John C. Langford

Published May 2, 2002• 9 citations
arXivHTMLPDF

Abstract

A generic approach for compiling any classical-block-compression algorithm (which works in bounded space and time) into a quantum-block-compression algorithm is presented. Using this technique, compression asymptotically approaching the von Neumann entropy of a qubit source can be achieved. The automatically compiled algorithms are competitive (in time and space complexity) with hand-constructed quantum-block-compression algorithms.

DOI
10.1103/physreva.65.052312

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A generic approach for compiling any classical-block-compression algorithm (which works in bounded space and time) into a quantum-block-compression algorithm is presented. Using this technique, compression asymptotically approaching the von Neumann entropy of a qubit source can be achieved. The automatically compiled algorithms are competitive (in time and space complexity) with hand-constructed quantum-block-compression algorithms.
Suppose that a finite dimensional quantum source is known to have von Neumann entropy less than or equal to SSS but is otherwise completely unspecified. We describe a method of universal quantum data compression which will faithfully compress the quantum information of any such source to SSS qubits per signal (in the limit of large block lengths).
Consider a source of pure quantum states with density matrix ρ and von Neumann entropy S . It is known that the quantum information of the source may be faithfully compressed to 5 qubits/signal (asymptotically for long sequences) and that this limit is optimal if we demand high fidelity for long strings of signals. The usual method for achieving this compression, based on the typical subspace of the source, requires knowledge of the density matrix. We describe a new “universal” compression scheme which does not even require knowledge of ρ. More precisely, suppose that the source is known to have von Neumann entropy less than some given bound S 0 but is otherwise completely unspecified. Our method will faithfully compress the quantum information of any such source to S 0 qubits/signal. It is known that S is always less than or equal to the Shannon entropy H of the prior distribution of the source. According to Shannon’s theorem, H provides the limit for compression of classical signals, so one may ask the question: what is the physical origin of the extra compression available in the quantum case compared to the classical case (where the prior probabilities are the same)? One may suggest that this effect is a consequence of possible non-orthogonality of signal states. However we demonstrate that it is possible to make each pair of signal states of a source more parallel (keeping the prior probabilities the same) while increasing the von Neumann entropy. This casts doubt on the conventional wisdom that overlap of quantum states is a measure of their (non-)distinguishability. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The accurate computation of advanced quantum algorithms like Shor’s integer factorization, quantum phase estimation (QPE), and the quantum Fourier transform (QFT) requires quantum circuits of considerable size and depth. It is difficult to achieve reliable computation with deep quantum circuits due to the limited coherence times of the current noisy quantum devices. The quantum fanout gate is known to be a powerful primitive for reducing the depth of many quantum circuits (Høyer and Špalek 2003; Gottesman and Chuang 1999). Shallow or constant-depth quantum circuits are desirable for both near-term and fault-tolerant quantum computations as they reduce noise and allow faster execution of quantum algorithms, potentially skirting the effects of short coherence times. In this work, we show new approaches towards implementation of quantum fanout gate. In particular, we show that by analogue time-evolving the quantum systems according to two well-studied Hamiltonians, namely, quantum Ising and quantum Heisenberg models, we can implement the quantum fanout operator using constantly many layers of digital quantum gates. Important foundations to the area of quantum encoding were provided by Schumacher who proved the quantum analog of Shannon’s noiseless coding theorem for an independent and identically distributed (i.i.d.) quantum source, (Schumacher 1995). In this work, we show a lossless, variable-length block encoding scheme of quantum information emitted from a completely general stochastic quantum source, and it is encoded into the Fock space. While doing so, we extend the notion of uniquely decodable (or completely lossless) quantum codes to be used for quantum block data compression. As our main result, for a fixed mlmany pure states emitted by a given quantum stochastic source, we derive the optimal lower bound of the average codeword length over a subset of uniquely decodable quantum codes called “special block codes,” which are applied to encode the pure states of m many blocks each of block size l. Additionally, we show that for quantum stationary sources in particular, the optimal lower bound of the average codeword length per symbol computed over a subset of special block codes called “constrained special block codes” equals the von Neumann entropy rate of the source for an asymptotically long block size.
We propose a series of quantum algorithms for computing a wide range of quantum entropies and distances, including the von Neumann entropy, quantum Rényi entropy, trace distance, and fidelity. The proposed algorithms significantly outperform the prior best (and even quantum) ones in the low-rank case, some of which achieve exponential speedups. In particular, forN-dimensional quantum states of rankr, our proposed quantum algorithms for computing the von Neumann entropy, trace distance and fidelity within additive error ε have time complexity of Õ(r/ε2), Õ(r5/ε6) and Õ(r6.5/ε7.5), respectively. By contrast, prior quantum algorithms for the von Neumann entropy and trace distance usually have time complexity Ω(N), and the prior best one for fidelity has time complexity Õ(r12.5/ε13.5). The key idea of our quantum algorithms is to extend block-encoding from unitary operators in previous work to quantum states (i.e., density operators). It is realized by developing several convenient techniques to manipulate quantum states and extract information from them. The advantage of our techniques over the existing methods is that no restrictions on density operators are required; in sharp contrast, the previous methods usually require a lower bound on the minimal non-zero eigenvalue of density operators.
We describe a universal information–compression scheme that compresses any pure quantum independent identically distributed (i.i.d.) source asymptotically to its von Neumann entropy, with no prior knowledge of the structure of the source. We introduce a diagonalization procedure that enables any classical compression algorithm to be used in a quantum context. Our scheme is then based on the corresponding quantum translation of the classical Lempel–Ziv algorithm. Our methods lead to a conceptually simple way of estimating the entropy of a source in terms of the measurement of an associated length parameter while maintaining high fidelity for long blocks. As a byproduct we also estimate the eigenbasis of the source. Since our scheme is based on the Lempel–Ziv method, it can potentially be applied also to non–i.i.d. sources that satisfy some further regularity conditions.