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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Computational synthesis of any n -qubit pure or mixed state

Song Lin, Colin P. Williams

Published Aug 5, 2003• 4 citations
Publisher

Abstract

Future quantum information processing devices will require the use of exotic quantum states, such as specially crafted entangled states, to achieve certain desired computations on demand. Thus far, synthesis schemes for such states have been devised on a case-by-case basis using ad hoc techniques. In this paper we present a systematic method for finding a quantum circuit that can synthesize any pure or mixed n-qubit state. We then give examples of the use of our algorithm for finding synthesis pathways for especially exotic quantum states such as maximal mixed states. It is not known how to prepare general instances of such states by other means. Thus our quantum state synthesis algorithm should be of use not only in quantum information processing, but also in experimental quantum physics.

DOI
10.1117/12.486347

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Future quantum information processing devices will require the use of exotic quantum states, such as specially crafted entangled states, to achieve certain desired computations on demand. Thus far, synthesis schemes for such states have been devised on a case-by-case basis using ad hoc techniques. In this paper we present a systematic method for finding a quantum circuit that can synthesize any pure or mixed n-qubit state. We then give examples of the use of our algorithm for finding synthesis pathways for especially exotic quantum states such as maximal mixed states. It is not known how to prepare general instances of such states by other means. Thus our quantum state synthesis algorithm should be of use not only in quantum information processing, but also in experimental quantum physics.
We study the problem of state synthesis in the DQC1 (One Clean Qubit) model of quantum computation, which provides a single pure qubit and nnn maximally mixed qubits, and after applying any quantum circuit some subset of the qubits are measured or discarded. In the case of discarding, we show that it is impossible to prepare additional pure qubits, and that it is impossible to prepare very low-temperature Gibbs states on additional qubits. In the case of measurements, we show that the probability of synthesizing mmm additional qubits is bounded by 21−m2^{1-m}21−m, and that the probability of preparing low-temperature Gibbs states is bounded by 22−m2^{2-m}22−m. As a consequence, we give a lower-bound the runtime of a recently studied class of repeated interaction quantum algorithms. The techniques used study states and circuits at the level of entries of their respective density and unitary matrices.
A common model of quantum computing is the gate model with binary basis states. Here, we consider the gate model of quantum computing with a non-binary radix resulting in more than two basis states to represent a quantum digit, or qudit. Quantum entanglement is an important phenomenon that is a critical component of quantum computation and communications algorithms. The generation and use of entanglement among radix-2 qubits is well-known and used often in quantum computing algorithms. Quantum entanglement exists in higher-radix systems as well although little is written regarding the generation of higher-radix entangled states. We provide background describing the feasibility of multiple-valued logic quantum systems and describe a new systematic method for generating maximally entangled states in quantum systems of dimension greater than two. This method is implemented in a synthesis algorithm that is described. Experimental results are included that demonstrate the transformations needed to create specific forms of maximally entangled quantum states.
This paper addresses the challenge of preparing mixed quantum states -- both arbitrary states in general and highly sparse ones in particular -- an area that has received far less attention than the preparation of pure states. We present two classes of circuit-synthesis methods: one based on constructing the density matrix as a mixture of pure states and the other based on purification. To improve both preprocessing efficiency and the complexity of the resulting circuits, we propose a novel strategy based on the Cholesky decomposition, which offers significant advantages, especially when the target density matrix is highly or extremely sparse. Furthermore, by exploiting incomplete Cholesky decomposition with threshold dropping, we introduce a practical approach for constructing high-fidelity approximations of the target density matrix. This approach enables substantial reductions in circuit complexity at the cost of negligible or only mild fidelity loss.
Quantum computation established itself as a promising emerging technology and, hence, attracted considerable attention in the domain of computer-aided-design (CAD). However, quantum mechanical phenomena such as superposition, phase shifts, or entanglement lead to a logic model which poses serious challenges to the development of a proper design flow for quantum circuits. Consequently, researchers addressed synthesis of quantum circuits not as a single design step, but considered sub-tasks such as synthesis of Boolean components or synthesis of restricted subsets of quantum functionality. Generating a particularly desired quantum state is another of these sub-tasks. However, logic synthesis of quantum circuits accomplishing that has hardly been considered yet. In this work, we propose a generic method which automatically synthesizes a quantum circuit generating any desired quantum state from an initially given basis state. The proposed method allows for both, a theoretical determination of upper bounds as well as an experimental evaluation of the number of quantum gates needed for this important design step.
Abstract The preparation of quantum states is a fundamental subroutine for a broad class of quantum information protocols and is critical for both quantum communication and quantum computation. Building upon the quantum algorithms introduced in previous works [M. B. Pozzobom and J. Maziero, Quantum Inf. Process. 18, 142 (2019)] and [E. R. Gårding et al., Entropy 23, 797 (2021)], the authors of [F. Shahbeigi, M. Karimi and V. Karimipour, Phys. Scr. 97, 025101 (2022)] demonstrated the capability to prepare mixed two-qubit X-real states on quantum computers by extending the methodology originally devised for mixed two-qubit Bell-diagonal states. In this article, we delve into an overlooked pattern within these quantum circuits, allowing us to present a modular algorithm for the preparation of general d -dimensional mixed quantum states using quantum information processors. Our general algorithm has a modular structure, encompassing eigenvalue encoding, entropy injection, and eigenvector preparation. To validate our algorithm, we conduct tests on quantum computers utilizing both X- and non-X-states for two mixed-state qubits, two-ququart Bell-diagonal states, as well as arbitrary random density matrices spanning one, two, and three qubits.
In order to demonstrate nontrivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 controlled-NOT (CNOT) gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requries at most 2 CNOT and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, yyy-rotation, zzz-rotation, and phase gates.