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arXiv:1006.3614 · Quantum Information and Computation

Metrics on unitary matrices and their application to quantifying the degree of non-commutativity between unitary matrices

H. F. Chau

Published Sep 1, 2011• 13 citations
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Abstract

By studying the minimum resources required to perform a unitary transformation, families of metrics and pseudo-metrics on unitary matrices that are closely related to a recently reported quantum speed limit by the author are found. Interestingly, this family of metrics can be naturally converted into useful indicators of the degree of non-commutativity between two unitary matrices.

DOI
10.26421/qic11.9-10-1

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By studying the minimum resources required to perform a unitary transformation, families of metrics and pseudo-metrics on unitary matrices that are closely related to a recently reported quantum speed limit by the author are found. Interestingly, this family of metrics can be naturally converted into useful indicators of the degree of non-commutativity between two unitary matrices.
Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] found a family of metrics and pseudo-metrics on nnn-dimensional unitary operators that can be interpreted as the minimum resources (given by certain tight quantum speed limit bounds) needed to transform one unitary operator to another. This result is closely related to the weighted ℓ1\ell^1ℓ1-norm on Rn{\mathbb R}^nRn. Here we generalize this finding by showing that every weighted ℓp\ell^pℓp-norm on Rn{\mathbb R}^nRn with $1\le p \le \limitingp$ induces a metric and a pseudo-metric on nnn-dimensional unitary operators with quantum information-theoretic meanings related to certain tight quantum speed limit bounds. Besides, we investigate how far the correspondence between the existence of metrics and pseudo-metrics of this type and the quantum speed limits can go.
Recently, Chau [Quant. Inform. & Comp. 11, 721 (2011)] showed that one can define certain metrics and pseudo-metrics on U(n), the group of all n×nn\times nn×n unitary matrices, based on the arguments of the eigenvalues of the unitary matrices. More importantly, these metrics and pseudo-metrics have quantum information theoretical meanings. So it is instructive to study this kind of metrics and pseudo-metrics on U(n). Here we show that any symmetric norm on Rn{\mathbb R}^nRn induces a metric on U(n). Furthermore, using the same technique, we prove an inequality concerning the eigenvalues of a product of two unitary matrices which generalizes a few inequalities obtained earlier by Chau [arXiv:1006.3614v1].
Quantum speed limits are relations yielding lower bounds on the evolution time of quantum systems. These results have been generalized in some ways, in particular by including evolutions to non-orthogonal states. However, there was a gap in the literature on this area, for only unitary evolutions -- closed quantum systems -- had been considered. On this Ph.D. thesis, such limitation is overcome: our main result is a bound for quantum-system evolutions in general, whether unitary or not, and correctly recovers the known bounds in the unitary case. Applications of this bound to several concrete cases of interest are herein presented. This bound is also used to extend to the non-unitary case the discussion of the role of entanglement in fast evolutions, leading to nontrivial results. For the derivation of the results, a geometric approach has been employed, which allows a clear interpretation of the bounds and a discussion of the criteria for their saturation. No previous knowledge of quantum-state geometry by the reader has been assumed.
Quantum speed limits are the bounds that define how quickly one quantum state can transform into another. Instead of focusing on the transformation between pairs of states, we provide bounds on the speed limit of quantum evolution by unitary operators in arbitrary dimensions. These do not depend on the initial and final state but depend only on the trace of the unitary operator that is to be implemented and the gross characteristics (average and variance) of the energy spectrum of the Hamiltonian which generates this unitary evolution. The bounds that we find can be thought of as the generalization of the Mandelstam-Tamm (MT) and the Margolus-Levitin (ML) bound for state transformations to implementations of unitary operators. We will discuss the application of these bounds in several classes of transformations that are of interest in quantum information processing.
Quantum computing is one of the most promising technologies of the near future, and the simulation of quantum many-body systems is a natural application. In this work, we present classical simulations of the ground states of light atomic nuclei within the p shell, from He 6 to B 10 , calculated within the nuclear shell model. We compare the performance of two leading variational quantum eigensolver algorithms: the unitary coupled cluster (UCC) and the adaptive derivative-assembled pseudo-Trotter (ADAPT) methods, introducing a new metric to quantify the use of quantum resources in each simulation. We find that Slater determinants are the most useful reference states for both approaches. Our analysis suggests that ADAPT is more efficient for nuclei close to magic numbers, while UCC tends to require fewer resources toward the mid shell. This work lays the groundwork for robust benchmarking of quantum algorithms in nuclear structure studies.