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Scientific Reports

Universal quantum simulation of single-qubit nonunitary operators using duality quantum algorithm

Chao Zheng

Published Feb 17, 2021• 54 citations
Publisher

Abstract

Quantum information processing enhances human's power to simulate nature in quantum level and solve complex problem efficiently. During the process, a series of operators is performed to evolve the system or undertake a computing task. In recent year, research interest in non-Hermitian quantum systems, dissipative-quantum systems and new quantum algorithms has greatly increased, which nonunitary operators take an important role in. In this work, we utilize the linear combination of unitaries technique for nonunitary dynamics on a single qubit to give explicit decompositions of the necessary unitaries, and simulate arbitrary time-dependent single-qubit nonunitary operator F(t) using duality quantum algorithm. We find that the successful probability is not only decided by F(t) and the initial state, but also is inversely proportional to the dimensions of the used ancillary Hilbert subspace. In a general case, the simulation can be achieved in both eight- and six-dimensional Hilbert spaces. In phase matching conditions, F(t) can be simulated by only two qubits. We illustrate our method by simulating typical non-Hermitian systems and single-qubit measurements. Our method can be extended to high-dimensional case, such as Abrams-Lloyd's two-qubit gate. By discussing the practicability, we expect applications and experimental implementations in the near future.

DOI
10.1038/s41598-021-83521-5

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Quantum information processing enhances human's power to simulate nature in quantum level and solve complex problem efficiently. During the process, a series of operators is performed to evolve the system or undertake a computing task. In recent year, research interest in non-Hermitian quantum systems, dissipative-quantum systems and new quantum algorithms has greatly increased, which nonunitary operators take an important role in. In this work, we utilize the linear combination of unitaries technique for nonunitary dynamics on a single qubit to give explicit decompositions of the necessary unitaries, and simulate arbitrary time-dependent single-qubit nonunitary operator F(t) using duality quantum algorithm. We find that the successful probability is not only decided by F(t) and the initial state, but also is inversely proportional to the dimensions of the used ancillary Hilbert subspace. In a general case, the simulation can be achieved in both eight- and six-dimensional Hilbert spaces. In phase matching conditions, F(t) can be simulated by only two qubits. We illustrate our method by simulating typical non-Hermitian systems and single-qubit measurements. Our method can be extended to high-dimensional case, such as Abrams-Lloyd's two-qubit gate. By discussing the practicability, we expect applications and experimental implementations in the near future.
Quantum channels describe general transformations of the state of a physical system and play a fundamental role in many fields. It promises wide range of applications, because any physical process can be represented as a quantum channel transforming an initial state into a final state. The evolution of system undergoing a quantum channel is not necessary unitary which makes it unrealisable without ancillary system. Inspired by the method performing non-unitary operator by the linear combination of unitary operations, we proposed a quantum algorithm for the simulation of universal single-qubit channel, described by a convex combination of two generalized extreme channels corresponding to four Kraus operators, and is scalable to arbitrary higher dimension. We demonstrate the whole algorithm experimentally using the universal IBM cloud quantum computer and study properties of different qubit quantum channels. We illustrate the quantum capacity of the general qubit quantum channels, which quantifies the amount of quantum information that can be protected. There is a general agreement between the theoretical predictions and the experiments, which strongly supported our method. By realizing arbitrary qubit channel, this work provides a universal way to explore various properties of quantum channel and novel prospect of quantum communication.
Different from the Hermitian case, non-Hermitian (NH) systems have novel properties and strongly relate to open and dissipative quantum systems. In this work, we investigate how to simulate τ-anti-pseudo-Hermitian systems in a Hermitian quantum device using linear combinations of unitaries and duality quantum algorithm. Specifying the τ to time-reversal (T) and parity-time-reversal (PT) operators, we construct the two NH two-level systems, design quantum circuits including three qubits, and decide the quantum gates explicitly in detail. We also calculate the success probabilities of the simulation. Experimental implementation can be expected in small quantum simulator.
Realizing nonunitary transformations on unitary-gate-based quantum devices is critically important for simulating a variety of physical problems, including open quantum systems and subnormalized quantum states. We present a dilation-based algorithm to simulate nonunitary operations using probabilistic quantum computing with only one ancilla qubit. We utilize the singular-value decomposition (SVD) to decompose any general quantum operator into a product of two unitary operators and a diagonal nonunitary operator, which we show can be implemented by a diagonal unitary operator in a one-qubit dilated space. While dilation techniques increase the number of qubits in the calculation, and thus the gate complexity, our algorithm limits the operations required in the dilated space to a diagonal unitary operator, which has known circuit decompositions. We use this algorithm to prepare random subnormalized two-level states on a quantum device with high fidelity. Furthermore, we present the accurate nonunitary dynamics of two-level open quantum systems in a dephasing channel and an amplitude-damping channel computed on a quantum device. The algorithm presented will be most useful for implementing general nonunitary operations when the SVD can be readily computed, which is the case for most operators in the noisy intermediate-scale quantum computing era.
Imaginary-time evolution is a powerful tool in the study of quantum physics. However, existing classical algorithms for simulating imaginary-time evolution suffer from high computational complexity as the dimension of the quantum system increases. In this study we propose a quantum algorithm for implementing imaginary-time evolution using nonunitary quantum circuits with one ancillary qubit. The success probability of our algorithm is a polynomial function of the output error and can be enhanced by reorganizing the terms of the Hamiltonian. To illustrate the practicality of our algorithm on current quantum devices, we conduct a demonstration on superconducting and trapped-ion quantum processors to calculate the ground-state energy and determine the most stable molecular structure of H2{\mathrm{H}}_{2}H2​. Additionally, we validate the feasibility of our algorithm by numerically simulating the ground-state energies of LiH molecules and the quantum Ising chain. In contrast to existing algorithms, our method provides a systematic approach to construct the required nonunitary circuits using universal quantum gates, making it suitable for experimental implementation. Our algorithm opens up possibilities for exploring other physical phenomena such as finite-temperature properties and non-Hermitian systems.
Electron transport in realistic physical and chemical systems often involves the nontrivial exchange of energy with a large environment, requiring the definition and treatment of open quantum systems. Because the time evolution of an open quantum system employs a nonunitary operator, the simulation of open quantum systems presents a challenge for universal quantum computers constructed from only unitary operators or gates. Here, we present a general algorithm for implementing the action of any nonunitary operator on an arbitrary state on a quantum device. We show that any quantum operator can be exactly decomposed as a linear combination of at most four unitary operators. We demonstrate this method on a two-level system in both zero and finite temperature amplitude damping channels. The results are in agreement with classical calculations, showing promise in simulating nonunitary operations on intermediate-term and future quantum devices.
\n\nIn this work, quantum simulation of the evolution of a two-level quantum system with a generalized anti-PT(APT)-symmetric Hamiltonian in arbitrary phase is investigated theoretically. The evolution is reinterpreted as a subsystem in a Hilbert space of higher dimensions, enabling it to construct any two-dimensional APT-symmetric Hamiltonian in a Hermitian quantum system. In some special phases, a two-qubit quantum computer is able to achieve the quantum simulation. However, a minimum six-dimensional Hilbert space is necessary for a general case, indicating that a qutrit is needed as an ancillary. We show how to realize the simulation process in a qubit-qutrit hybrid system using the duality quantum algorithm. We also give the flow chart and quantum circuit for a quantum computer of three qubits to realize the quantum simulation. Therefore, experimental realizations and applications can be expected in the future.\n