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arXiv:2207.10007 · Quantum

Two-Unitary Decomposition Algorithm and Open Quantum System Simulation

Nishchay Suri, Joseph Barreto, Stuart Hadfield, Nathan Wiebe, Filip Wudarski, Jeffrey Marshall

Published May 15, 2023• 34 citations
arXivHTMLPDF

Abstract

Simulating general quantum processes that describe realistic interactions of quantum systems following a non-unitary evolution is challenging for conventional quantum computers that directly implement unitary gates. We analyze complexities for promising methods such as the Sz.-Nagy dilation and linear combination of unitaries that can simulate open systems by the probabilistic realization of non-unitary operators, requiring multiple calls to both the encoding and state preparation oracles. We propose a quantum two-unitary decomposition (TUD) algorithm to decompose a d -dimensional operator A with non-zero singular values as A=(U1+U2)/2 using the quantum singular value transformation algorithm, avoiding classically expensive singular value decomposition (SVD) with an O(d3) overhead in time. The two unitaries can be deterministically implemented, thus requiring only a single call to the state preparation oracle for each. The calls to the encoding oracle can also be reduced significantly at the expense of an acceptable error in measurements. Since the TUD method can be used to implement non-unitary operators as only two unitaries, it also has potential applications in linear algebra and quantum machine learning.

DOI
10.22331/q-2023-05-15-1002

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Simulating general quantum processes that describe realistic interactions of quantum systems following a non-unitary evolution is challenging for conventional quantum computers that directly implement unitary gates. We analyze complexities for promising methods such as the Sz.-Nagy dilation and linear combination of unitaries that can simulate open systems by the probabilistic realization of non-unitary operators, requiring multiple calls to both the encoding and state preparation oracles. We propose a quantum two-unitary decomposition (TUD) algorithm to decompose a d -dimensional operator A with non-zero singular values as A=(U1+U2)/2 using the quantum singular value transformation algorithm, avoiding classically expensive singular value decomposition (SVD) with an O(d3) overhead in time. The two unitaries can be deterministically implemented, thus requiring only a single call to the state preparation oracle for each. The calls to the encoding oracle can also be reduced significantly at the expense of an acceptable error in measurements. Since the TUD method can be used to implement non-unitary operators as only two unitaries, it also has potential applications in linear algebra and quantum machine learning.
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.
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.
One of the promises of quantum computing is to simulate physical systems efficiently. However, the simulation of open quantum systems - where interactions with the environment play a crucial role - remains challenging for quantum computing, as it is impossible to implement deterministically non-unitary operators on a quantum computer without auxiliary qubits. The Stinespring dilation can simulate an open dynamic but requires a high circuit depth, which is impractical for NISQ devices. An alternative approach is parallel probabilistic block-encoding methods, such as the Sz.-Nagy and Singular Value Decomposition dilations. These methods result in shallower circuits but are hybrid methods, and we do not simulate the quantum dynamic on the quantum computer. In this work, we describe a divide-and-conquer strategy for preparing mixed states to combine the output of each Kraus operator dilation and obtain the complete dynamic on quantum hardware with a lower circuit depth. The work also introduces a balanced strategy that groups the original Kraus operators into an expanded operator, leading to a trade-off between circuit depth, CNOT count, and number of qubits. We perform a computational analysis to demonstrate the advantages of the new method and present a proof-of-concept simulation of the Fenna-Matthews-Olson dynamic on current quantum hardware.
Singular Value Decomposition (SVD) is one of the most useful techniques for analyzing data in linear algebra. SVD decomposes a rectangular real or complex matrix into two orthogonal matrices and one diagonal matrix. The proposed Quantum-SVD algorithm interpolates the non-uniform angles in the Fourier domain. The error of the Quantum-SVD approach is some orders lower than the error given by ordinary Quantum Fourier Transformation. Our Quantum-SVD algorithm is a fundamentally novel approach for the computation of the Quantum Fourier Transformation (QFT) of non-uniform states. The presented Quantum-SVD algorithm is based on the singular value decomposition mechanism, and the computation of Quantum Fourier Transformation of non-uniform angles of a quantum system. The Quantum-SVD approach provides advantages in terms of computational structure, being based on QFT and multiplications.
Nonautonomous linear ordinary differential equations of the form v˙(t)=A(t) v(t)\dot{v}(t) = A(t)\, v(t)v˙(t)=A(t)v(t), where A(t)A(t)A(t) is non-skew-symmetric, are often used to describe nonunitary dynamics in a variety of fields that range from open quantum system dynamics to economic modeling. Because quantum computing hardware is designed to natively implement unitary transformations, existing algorithms for solving such equations on quantum hardware are based on the assumption that the nonunitary propagator is known, and use dilation techniques to embed the nonunitary dynamics within the unitary dynamics of a larger system. However, with the exception of cases where the nonunitary propagator is known in closed form, it needs to be calculated and manipulated on a classical computer at each time step. In this paper, we propose a quantum algorithm that does not require a priori knowledge of the explicit nonunitary propagator and effectively performs the dilation on the quantum hardware. Our algorithm combines a dilation scheme that uses singular value decomposition (SVD) to write the nonunitary propagator as a sum of unitaries with simulating the dynamics of the SVD factors on the quantum hardware. The population-only time-convolutionless quantum master equation describing photoinduced charge transfer in a solvated molecular triad is used as a demonstrative example of the applicability of the algorithm and its sensitivity to noise.
In quantum computing, every operation must be unitary to ensure reversibility and preserve probability amplitudes, which is a fundamental requirement for any quantum process. However, many real-world computations in areas such as machine learning, signal processing, and dynamical modeling inherently involve non-unitary matrices. Since these cannot be directly implemented on quantum hardware, a practical workaround is required. One promising solution is block-encoding, a framework that embeds a non-unitary matrix A in C^{n x n} into a larger unitary matrix U_A in C^{2n x 2n}, enabling the simulation of non-unitary actions within quantum circuits through ancilla preparation and post-selection. This thesis presents a novel and efficient method for block-encoding based on Gram-Schmidt orthogonalization. The algorithm constructs a unitary U_A with A embedded as the top-left block and remaining components filled to ensure global unitarity. Compared to traditional Singular Value Decomposition (SVD)-based approaches, the proposed method achieves at least 20% improvement in execution time, without compromising output fidelity or numerical stability. A corresponding quantum circuit is developed to extract A|psi> (that is, the transformed version of the input quantum state |psi> after applying the linear operator A) using post-selection on ancilla qubits, with the success probability determined by