New Quantum Algorithms for Computing Quantum Entropies and Distances

Abstract

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 best known (and even quantum) ones in the low-rank case, some of which achieve exponential speedups. In particular, for Ndimensional quantum states of rank r, our proposed quantum algorithms for computing the von Neumann entropy, trace distance and fidelity within additive error ε have time complexity of Õ r 2 /ε 2 , Õ r 5 /ε 6 and Õ r 6.5 /ε 7.5 1 , respectively. In contrast, the known algorithms for the von Neumann entropy and trace distance require quantum time complexity of Ω(N ) [AISW19,GL20,GHS21], and the best known one for fidelity requires Õ r 21.5 /ε 23.5 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. In particular, we introduce a novel technique for eigenvalue transformation of density operators and their (non-integer) positive powers, based on the powerful quantum singular value transformation (QSVT) [GSLW19]. 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 of the minimal non-zero eigenvalue of density operators. In addition, we provide some techniques of independent interest for trace estimation, linear combinations, and eigenvalue threshold projectors of (subnormalized) density operators, which will be, we believe, useful in other quantum algorithms.

Quantum Phase Processing: Transform and Extract Eigen-Information of Quantum Systems

“…For example, f (x) = x is just a simple one-term polynomial, whereas f (x) = √ x takes much more terms to precisely approximate. Thus such a method presented in [48] may lead to even worse complexity than previous ones in [11][12][13]. We defer further discussion and comparison to Appendix E 4.…”

“…The polynomial transformation implemented by QSVT lies in amplitudes of the outcome state, which could not be obtained by indirect measurements of the ancilla qubit in QSVT, thus most QSVT-based entropies estimation algorithms estimate the value either by applying amplitude estimation or combining with the DQC1 model, and both of these methods increase the circuit size. Another approach is using a polynomial to estimate the square root of the function f (x), as shown by Wang et al [48]. This approach makes QSVT compatible with indirect measurements, since the approximated function now can be represented by the probability of measuring, which is similar as in QPP.…”

“…We present further comparison on different entropies estimation algorithms in this section. In Table S1, we add the von Neumann entropy estimation algorithm proposed by Wang et al [48]. When assuming rank κ, the QPP-based algorithm improves the result in [48] by a factor of κ.…”

“…In Table S1, we add the von Neumann entropy estimation algorithm proposed by Wang et al [48]. When assuming rank κ, the QPP-based algorithm improves the result in [48] by a factor of κ. In Table S2, we add Rényi entropy estimation algorithms proposed by Wang et al [48].…”

“…When assuming rank κ, the QPP-based algorithm improves the result in [48] by a factor of κ. In Table S2, we add Rényi entropy estimation algorithms proposed by Wang et al [48]. Here the O notation omits the log and α factors.…”