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arXiv:quant-ph/0608196 · Physical Review A

Visualization of superposition of macroscopically distinct states

Tomoyuki Morimae, Akira Shimizu

Published Nov 22, 2006• 9 citations
arXivHTMLPDF

Abstract

We propose a method of visualizing superpositions of macroscopically distinct states in many-body pure states. We introduce a visualization function, which is a coarse-grained quasi joint probability density for two or more Hermitian additive operators. If a state contains superpositions of macroscopically distinct states, one can visualize them by plotting the visualization function for appropriately taken operators. We also explain how to efficiently find appropriate operators for a given state. As examples, we visualize four states containing superpositions of macroscopically distinct states: the ground state of the $XY$ model, that of the Heisenberg antiferromagnet, a state in Shor's factoring algorithm, and a state in Grover's quantum search algorithm. Although the visualization function can take negative values, it becomes nonnegative (hence, becomes a coarse-grained joint probability density) if the characteristic width of the coarse-graining function used in the visualization function is sufficiently large.

DOI
10.1103/physreva.74.052111

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We propose a method of visualizing superpositions of macroscopically distinct states in many-body pure states. We introduce a visualization function, which is a coarse-grained quasi joint probability density for two or more Hermitian additive operators. If a state contains superpositions of macroscopically distinct states, one can visualize them by plotting the visualization function for appropriately taken operators. We also explain how to efficiently find appropriate operators for a given state. As examples, we visualize four states containing superpositions of macroscopically distinct states: the ground state of the XYXYXY model, that of the Heisenberg antiferromagnet, a state in Shor's factoring algorithm, and a state in Grover's quantum search algorithm. Although the visualization function can take negative values, it becomes nonnegative (hence, becomes a coarse-grained joint probability density) if the characteristic width of the coarse-graining function used in the visualization function is sufficiently large.
We consider a coarse‐grained quasi joint probability density Ξ of two macroscopic observables. If a quantum many‐body state has a superposition of macroscopically distinct states, the structure of the superposition can be visualized by plotting Ξ for appropriate macroscopic observables. For example, the ground state of the XY model, that of the Heisenberg antiferromagnet, and a state in Grover's quantum search algorithm, which have superpositions of macroscopically distinct states, can be visualized. Like the Wigner distribution function, Ξ can take negative values. However, if the characteristic width W of the coarse‐graining is sufficiently large, Ξ becomes nonnegative. We study a sufficient magnitude of such W. We see that W = O(N) is not sufficient, but W = O(N) would be sufficient, where N is the size of the system.
For quantum computation, we investigate the conjecture that the superposition of macroscopically distinct states is necessary for a large quantum speedup. Although this conjecture was supported for a circuit-based quantum computer performing Shor's factoring algorithm [A. Ukena and A. Shimizu, Phys. Rev. A69 (2004) 022301], it needs to be generalized for it to be applicable to a large class of algorithms and/or other models such as measurement-based quantum computers. To treat such general cases, we first generalize the indices for the superposition of macroscopically distinct states. We then generalize the conjecture, using the generalized indices, in such a way that it is unambiguously applicable to general models if a quantum algorithm achieves exponential speedup. On the basis of this generalized conjecture, we further extend the conjecture to Grover's quantum search algorithm, whose speedup is large but quadratic. It is shown that this extended conjecture is also correct. Since Grover's algorithm is a representative algorithm for unstructured problems, the present result further supports the conjecture.
We investigate macroscopic entanglement of quantum states in quantum computers, where we say a quantum state is entangled macroscopically if the state has superposition of macroscopically distinct states. The index ppp of the macroscopic entanglement is calculated as a function of the step of the computation, for Grover's quantum search algorithm and Shor's factoring algorithm. It is found that whether macroscopically entangled states are used or not depends on the numbers and properties of the solutions to the problem to be solved. When the solutions are such that the problem becomes hard in the sense that classical algorithms take more than polynomial steps to find a solution, macroscopically entangled states are always used in Grover's algorithm and almost always used in Shor's algorithm. Since they are representative algorithms for unstructured and structured problems, respectively, our results support strongly the conjecture that quantum computers utilize macroscopically entangled states when they solve hard problems much faster than any classical algorithms.
What are the conditions for adiabatic quantum computation (AQC) to outperform classical computation? Although there exist several quantum adiabatic algorithms achieving the strong quantum speedup, the essential keys to their speedups are still unclear. Here, we investigate the connection between superpositions of macroscopically distinct states and known examples of the speedup in AQC. To formalize this notion we consider an index ppp that quantifies a superposition of macroscopically distinct states from the asymptotic behaviors of fluctuations of additive observables. We determine this index for five examples of adiabatic algorithms exhibiting various degrees of the speedup. The results suggest that the superposition of macroscopically distinct states is an appropriate indicator of entanglement crucial to the strong quantum speedup in AQC.
This paper presents an introduction to geometric representations of quantum states in which each distinct quantum state, pure and mixed, corresponds to a unique point in a Euclidean space. Beginning with a review of some underappreciated properties of the most commonly used geometric representation, the Bloch sphere visualization of qubit states, we show how concepts, algorithms, and spatial relations viewable on this geometric representation can be extended to representations of qudit states of any finite quantum dimension ddd and on to the infinite-dimensional limit. A primary goal of the work is helping the reader develop a visual intuition of these spaces, which can complement the understanding of the algebraic formalism of quantum mechanics for learners, teachers, and researchers at any level. Particular emphasis is given both to understanding states in a basis-independent way and to understanding how probability amplitudes and density matrix elements used to algebraically represent states in a particular basis correspond to line segments and angles in the geometric representations. In addition to providing visualizations for such concepts as superpositions, mixtures, decoherence, and measurement, we demonstrate how the representations can be used to substitute simple geometrical calculations for more cumbersome linear algebra ones, which may be of particular use in introducing mixed states and density matrices to beginning quantum students at an early stage. The work concludes with the geometrical interpretation of some commonly used metrics such as the purity of states and their relation to real, Euclidean vectors in the infinite-dimensional limit of the space, which contains all lower-dimensional qudit spaces as subspaces.
What are the conditions for adiabatic quantum computation (AQC) to outperform classical computation? Although there exist several quantum adiabatic algorithms achieving the strong quantum speedup, the essential keys to their speedups are still unclear. Here, we investigate the connection between superpositions of macroscopically distinct states and known examples of the speedup in AQC. To formalize this notion we consider an index ppp that quantifies a superposition of macroscopically distinct states from the asymptotic behaviors of fluctuations of additive observables. We determine this index for five examples of adiabatic algorithms exhibiting various degrees of the speedup. The results suggest that the superposition of macroscopically distinct states is an appropriate indicator of entanglement crucial to the strong quantum speedup in AQC.