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arXiv:quant-ph/0310126 · Physical Review A

Quantum algorithms for phase-space tomography

Juan Pablo Paz, Augusto J. Roncaglia, Marcos Saraceno

Published Mar 18, 2004• 27 citations
arXivHTMLPDF

Abstract

We present efficient circuits that can be used for the phase-space tomography of quantum states. The circuits evaluate individual values or selected averages of the Wigner, Kirkwood, and Husimi distributions. These quantum gate arrays can be programmed by initializing appropriate computational states. The Husimi circuit relies on a subroutine that is also interesting in its own right: the efficient preparation of a coherent state, which is the ground state of the Harper Hamiltonian.

DOI
10.1103/physreva.69.032312

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We present efficient circuits that can be used for the phase-space tomography of quantum states. The circuits evaluate individual values or selected averages of the Wigner, Kirkwood, and Husimi distributions. These quantum gate arrays can be programmed by initializing appropriate computational states. The Husimi circuit relies on a subroutine that is also interesting in its own right: the efficient preparation of a coherent state, which is the ground state of the Harper Hamiltonian.
A programmable gate array is a circuit whose action is controlled by input data. In this paper we describe a special-purpose quantum circuit that can be programmed to evaluate the expectation value of any operator O acting on a space of states of N dimensions. The circuit has a program register whose state $|\ensuremath{\Psi}(O){〉}_{P}$ encodes the operator O whose expectation value is to be evaluated. The method requires knowledge of the expansion of O in a basis of the space of operators. We discuss some applications of this circuit and its relation to known instances of quantum state tomography.
We study the efficiency of quantum algorithms which aim at obtaining phase-space distribution functions of quantum systems. Wigner and Husimi functions are considered. Different quantum algorithms are envisioned to build these functions, and compared with the classical computation. Different procedures to extract more efficiently information from the final wave function of these algorithms are studied, including coarse-grained measurements, amplitude amplification, and measure of wavelet-transformed wave function. The algorithms are analyzed and numerically tested on a complex quantum system showing different behavior depending on parameters: namely, the kicked rotator. The results for the Wigner function show in particular that the use of the quantum wavelet transform gives a polynomial gain over classical computation. For the Husimi distribution, the gain is much larger than for the Wigner function and is larger with the help of amplitude amplification and wavelet transforms. We discuss the generalization of these results to the simulation of other quantum systems. We also apply the same set of techniques to the analysis of real images. The results show that the use of the quantum wavelet transform allows one to lower dramatically the number of measurements needed, but at the cost of a large loss of information.
We present a method that outputs a sequence of simple unitary operations to prepare a given quantum state that is a generalized coherent state. Our method takes as inputs the expectation values of some relevant observables. Such expectations can be estimated by performing projective measurements on O(M3⁡log(M/δ)/ε2) copies of the state, where M is the dimension of an associated Lie algebra, ɛ is a precision parameter, and 1 − δ is the required confidence level. The method can be implemented on a classical computer and runs in time O(M4⁡log(M/ε)). It provides O(M⁡log(M/ε)) simple unitaries that form the sequence. The overall complexity is then polynomial in M, being very efficient in cases where M is significantly smaller than the Hilbert space dimension, as for some fermion algebras. When the algebra of relevant observables is given by certain Pauli matrices, each simple unitary may be easily decomposed into two-qubit gates. We discuss applications to efficient quantum state tomography and classical simulations of quantum circuits.
The quasiprobability distribution of the discrete Wigner function provides a complete description of a quantum state and is, therefore, a useful alternative to the usual density matrix description. Moreover, the experimental quantum state tomography in discrete Wigner phase space can also be implemented. We observe that for a certain class of states, such as harmonic states, the Wigner matrix is far more sparse compared to the density matrix in the computational basis. Additionally, reading only a small part of the Wigner matrix may suffice to infer certain behavior of quantum dynamics. In such cases, selective Wigner phase space tomography (SWPST) can be more efficient than the usual density matrix tomography (DMT). Employing nuclear magnetic resonance methods on a three-qubit nuclear spin register, we experimentally estimate Wigner matrices of various two-qubit quantum states. As a specific example application of SWPST, we study the evolution of spin coherent states under the quantum chaotic kicked top model and extract signatures of quantum-classical correspondence in the Wigner phase space.
Gaussian states hold a fundamental place in quantum mechanics, quantum information, and quantum computing. Many subfields, including quantum simulation of continuousvariable systems, quantum chemistry, and quantum machine learning, rely on the ability to accurately and efficiently prepare states that reflect a Gaussian profile in their probability amplitudes. Although Gaussian states are natural in continuousvariable systems, the practical interest in digital, gate-based quantum computers demands discrete approximations of Gaussian distributions over a computational basis of size2n2^{n}2n. Because of the exponential scaling of naive amplitude-encoding approaches and the cost of certain block-encoding or Hamiltonian simulation techniques, a resource-efficient preparation of approximate Gaussian states is required. In this work, we propose and analyze a circuit-based approach that starts with singlequbit rotations to form an exponential amplitude profile and then applies the quantum Fourier transform to map those amplitudes into an approximate Gaussian distribution. We demonstrate that this procedure achieves high fidelity with the target Gaussian state while allowing optional pruning of small controlled-phase angles in the quantum Fourier transform, thus reducing gate complexity to near-linear inO(n)\mathcal{O}(n)O(n). We conclude that the proposed technique is a promising route to make Gaussian states accessible on noisy quantum hardware and to pave the way for scalable implementations on future devices. The implementation of this algorithm is available at the Classiq library: https://github.com/classiq/classiq-library.
Preparing complex quantum states is of limited value without efficient methods for extracting useful classical information from them. This thesis studies when such information can be extracted efficiently, given copies of an unknown quantum state or query access to an unknown unitary transformation. We begin with pure state tomography. We present an algorithm that uses only nonadaptive single-qubit Pauli measurements yet achieves optimal copy and time complexities up to logarithmic factors. This is the first full state tomography algorithm with near-optimal running time. We then show how additional structure can make tomography time-efficient, giving algorithms for two structured classes of quantum states: non-interacting fermions and near-stabilizer states. We also introduce a new learning task, agnostic tomography, where the goal is to find the best approximation to an arbitrary quantum state from a structured class of states, and give efficient algorithms in this model. Turning to property testing, we give an efficient tester for stabilizer dimension, which implies that Ω(n) T-gates are necessary for any Clifford+T circuit to prepare n-qubit computationally pseudorandom states. We also prove copy-complexity lower bounds for testing low-degree phase states, showing that testing is nearly as hard as learning these states. More strongly, for every fixed degree d, ensembles of n-qubit degree-d phase states form exponentially accurate approximate state t-designs for t=Ω(n^{⌊(d-1)/2⌋}). Finally, we extend these questions from states to unitary transformations. We develop a general framework for learning unitary transformations by exploiting low-complexity Pauli spectra. This framework unifies and extends prior learning algorithms for Clifford and near-Clifford circuits, shallow circuits, quantum juntas, and fermionic matchgate circuits.