Quantum Simulation Achieves Imaginary Time Evolution Without Classical Computation.

The pursuit of understanding the lowest energy state, or ground state, of complex quantum systems remains a central challenge in physics and chemistry. Traditional computational methods often struggle with the exponential increase in complexity as system size grows. Researchers now present a novel approach to simulating imaginary-time evolution, a technique used to efficiently determine these ground states, circumventing the limitations of conventional methods by eliminating the need for computationally intensive intermediate calculations. S. Alipour and T. Ojanen, both from the Computational Physics Laboratory at Tampere University, detail their findings in a paper entitled “State-Based Quantum Simulation of Imaginary-Time Evolution”, where they demonstrate how controlled-SWAP gates and measurements, utilising a state-based quantum computing technique, can effectively simulate this non-unitary process. This state-based approach, differing from conventional quantum computation which relies on sequences of quantum gates, employs a broader range of quantum states to represent and manipulate information. A controlled-SWAP gate exchanges the states of two qubits based on the state of a control qubit, and is a fundamental operation in quantum information processing.

Quantum simulations currently face significant challenges in determining the lowest energy state, or ground state, of complex systems. Traditional methods rely on imaginary time evolution, a process that mimics the system’s behaviour over an extended, imaginary timescale to converge towards the ground state. However, implementing this on quantum computers necessitates intricate sequences of quantum gates and repeated measurements of the system’s state, creating computational bottlenecks and limiting scalability. Recent research details a novel approach that circumvents these limitations by representing quantum evolution directly through a carefully constructed set of quantum states, rather than relying on gate-based operations.

This state-based method utilises controlled-SWAP gates—quantum gates that exchange the states of two qubits—and projective measurements to simulate non-unitary imaginary time evolution directly on a quantum processor. This eliminates the need for intermediate classical computation and the demanding task of full state tomography, a process of reconstructing the quantum state from a series of measurements. The ability to simulate non-unitary evolution expands the range of accessible quantum simulations, potentially accelerating progress in fields such as materials science, drug discovery, and fundamental physics.

Researchers demonstrate the efficacy of their approach by applying it to the Ising model, a foundational model in condensed matter physics. The Ising model describes the interactions between magnetic spins and is used to understand phenomena including magnetism and phase transitions. By employing the state-based method, they accurately determine the ground state of the Ising model, validating the technique and opening avenues for exploring more complex quantum systems.

The research also establishes rigorous bounds on the accuracy of variational quantum eigensolver (VQE) simulations, a hybrid quantum-classical algorithm used to approximate the ground state of quantum systems. These bounds, applied specifically to the Ising model in a transverse field, reveal a clear relationship between simulation parameters and the resulting error in the estimated ground state. The number of qubits employed and the duration of the simulation directly influence accuracy, with larger energy gaps—the difference in energy between the ground state and the first excited state—facilitating more accurate estimations. Crucially, the initial fidelity—a measure of how closely the initial quantum state resembles the true ground state—plays a significant role in determining the overall error.

The analysis centres on quantifying the distance between the state estimated by VQE and the true ground state, utilising the Bures distance—a metric for comparing quantum states. This provides explicit relationships between simulation parameters, such as the number of qubits, simulation time, and initial fidelity, and the resulting error. The derived bounds incorporate the energy gap of the Hamiltonian, confirming that larger gaps lead to more accurate estimations.

Researchers are investigating the scalability of this state-based imaginary time evolution as system size increases. Combining these theoretical bounds with adaptive VQE strategies—where simulation parameters are adjusted during the optimisation process—holds the potential for more efficient and accurate quantum simulations. Empirical validation of these bounds through numerical simulations and, ultimately, on quantum hardware, is essential to confirm their practical relevance and ensure the technique’s viability for tackling increasingly complex quantum systems.

Future work will focus on combining these theoretical bounds with adaptive VQE strategies, dynamically adjusting simulation parameters during the optimisation process to enhance efficiency and accuracy. Researchers plan to validate these bounds through numerical simulations and, crucially, on quantum hardware, confirming the technique’s potential for addressing increasingly complex quantum systems.