Quantum Simulations Become Far More Efficient with New Algorithm

Joshua M. Courtney, University of Georgia, presents a new method for simulating non-Hermitian Hamiltonians, a key step towards modelling complex physical systems beyond those described by traditional quantum mechanics. The method uses a bivariate extension of quantum signal processing, achieving query-optimal simulations with a complexity of $\mathcal{O}((α_R + β_I’T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$, matching established information-theoretic limits. This advancement overcomes limitations in simulating non-Hermitian systems by encoding the Dyson series as a polynomial and deterministically calculating angles via classical precomputation, offering improved efficiency and scalability for quantum simulations.

Compressing non-Hermitian dynamics via Dyson series polynomial approximation and block encoding

Alistair Hay and colleagues developed a quantum simulation technique for non-Hermitian Hamiltonians, utilising block encoding to compress complex operations. This compression is key, as simulating non-Hermitian systems traditionally demands significant computational resources that scale with multiple properties of the modelled system. Non-Hermitian Hamiltonians, unlike their Hermitian counterparts, do not guarantee real eigenvalues, necessitating different approaches to ensure stable and meaningful simulation results. The algorithm accesses Hamiltonian components, $H_R$ and $H_I$, through independent block encodings, requiring a constant overhead of two qubits regardless of system size. Block encoding represents a crucial technique in quantum computation, allowing complex unitary operations to be efficiently represented and implemented on a quantum computer using a smaller number of gates. They achieve this by mapping the unitary operation onto a higher-dimensional Hilbert space, effectively compressing the computational requirements.

Classical pre-computation determines angles in O(dR⋅dI) operations, where dR and dI represent properties of $H_R$ and $H_I$ respectively. These properties relate to the dimensions of the Hilbert spaces associated with each Hamiltonian component, influencing the computational cost of angle determination. Utilising block encoding allowed the system’s evolution, described by the Dyson series, a mathematical series breaking down complex changes over time into smaller steps, to be represented as a polynomial within this compressed circuit. The Dyson series provides a perturbative expansion for the time evolution operator under a non-Hermitian Hamiltonian, allowing for the approximation of the system’s dynamics. This new approach achieves a query complexity of O((αR + βI)T + log(1/ε)/log(1/ε)), surpassing previous methods by enabling additive scaling in operator norms and improving precision. The query complexity represents the number of times the algorithm needs to access the Hamiltonian during the simulation, a critical metric for evaluating its efficiency. Additive scaling in operator norms signifies that the computational cost increases linearly with the magnitude of the Hamiltonian’s components, offering a significant advantage over methods with multiplicative scaling.

The technique utilises a bivariate extension of quantum signal processing, offering a substantial advantage for complex quantum systems, and crosses a long-standing barrier in quantum simulation. Previously, efficient modelling of systems where Hermitian and non-Hermitian components contributed multiplicatively to computational cost was prevented. Quantum signal processing is a powerful technique for implementing phase estimation and related algorithms, and its bivariate extension allows for the efficient simulation of time evolution under non-Hermitian Hamiltonians. The algorithm calculates the angles required for the simulation classically beforehand in O(dR · dI) operations, with dR and dI representing the dimensions associated with each Hamiltonian. This classical pre-computation step is crucial for reducing the quantum circuit complexity and improving the overall efficiency of the algorithm. The resulting angles are then used to control the quantum gates in the simulation circuit.

Achieving scalable quantum simulation via optimised query complexity for non-Hermitian systems

The success probability of the algorithm is determined by both an intrinsic factor linked to the initial state, and a block-encoding factor dependent on the polynomial walk operator, resulting in a probability of e−2βI T∥e−iHeffT |ψ0⟩∥2. This probability indicates the likelihood of obtaining a correct simulation result, and is influenced by the accuracy of the block encoding and the initial state of the system. Achieving query optimality, requiring the fewest possible interactions with the system’s data, represents a substantial step forward for modelling complex, non-Hermitian systems. Query optimality is a fundamental goal in quantum algorithm design, as it minimises the number of expensive operations required to obtain a result. This advance builds upon a bivariate extension of quantum signal processing, a method for encoding quantum evolution as a polynomial. The bitorus, a mathematical surface formed by the Cartesian product of two circles, provides a convenient framework for representing the polynomial encoding of the Dyson series.

Quantum simulation efficiency is linked to initial system state

However, the algorithm’s reliance on block encoding introduces a practical tension; while theoretically efficient, the compression’s success is heavily influenced by the initial state of the system being modelled. The initial state of the system significantly impacts how well the compression works, introducing a variable element into the process. Specifically, the overlap between the initial state and the encoded subspace affects the fidelity of the simulation. Simulating complex systems often demands pushing the boundaries of computational feasibility, and this new technique for non-Hermitian Hamiltonians represents a genuine advance in that endeavour. The ability to efficiently simulate non-Hermitian systems has implications for a wide range of fields, including open quantum systems, quantum chemistry, and the study of exceptional points in parameter space.

These systems defy traditional modelling due to their unique mathematical properties. A new method for simulating non-Hermitian Hamiltonians, complex systems vital for modelling phenomena beyond standard quantum mechanics, has been established. Extending quantum signal processing encoded the evolution of these systems as a mathematical polynomial, allowing for deterministic calculation of important simulation angles using conventional computing. Scaling operator norms additively and improving simulation precision represents a major advance, opening avenues for modelling previously intractable systems. The potential applications extend to areas such as modelling the decay of unstable particles, simulating the dynamics of driven-dissipative quantum systems, and exploring the behaviour of systems with parity-time symmetry. Further research will focus on mitigating the sensitivity to the initial state and exploring the limits of scalability for even more complex non-Hermitian systems.

The researchers successfully achieved query-optimal quantum simulations of non-Hermitian Hamiltonians using an extension of quantum signal processing. This development matters because it provides a more efficient method for modelling complex systems that do not adhere to the rules of standard quantum mechanics, such as those found in open quantum systems and quantum chemistry. The algorithm encodes system evolution as a polynomial, enabling deterministic angle calculation and scaling operator norms additively, with a query complexity matching a known information-theoretic lower bound. The authors intend to address the algorithm’s sensitivity to the initial system state and explore scalability for increasingly complex systems.