Quantum Circuits Unlock New Ways to Simulate Complex Magnetic Materials

Researchers are increasingly exploring the connections between bosonic counting problems and the simulation of complex quantum systems. Minhyeok Kang from SKKU Advanced Institute of Nanotechnology, Gwonhak Lee from the same institution, and Youngrong Lim from Chungbuk National University, alongside Joonsuk Huh et al., demonstrate a significant advance in this field by extending the established link between Ising model interactions and matrix functions to encompass arbitrary network topologies. Their work reveals that transition amplitudes within the Ising Hamiltonian correlate directly with the hafnian and loop-hafnian, functions previously associated with classical computational hardness. Importantly, this research establishes a unified framework connecting single photons, Gaussian states, and spin dynamics, potentially unlocking new applications for quantum circuit models and highlighting previously unrecognised classically intractable problems.

This work extends the Ising model construction to arbitrary interaction networks, demonstrating that transition amplitudes of the Ising Hamiltonian are proportional to the hafnian and the loop-hafnian.

These matrix functions, including the permanent, hafnian, and loop-hafnian, are central to bosonic counting problems and arise naturally in spin models like the Ising and Heisenberg models. The correspondence between these seemingly disparate areas has implications for understanding the classical hardness of simulating interacting spin systems, relating their output distributions to computationally challenging #P-hard quantities.
Previous investigations largely focused on bipartite spin interactions, where transition amplitudes were proportional to the permanent. This research broadens this scope by showing that arbitrary interaction networks within the Ising model yield transition amplitudes proportional to both the hafnian and the loop-hafnian.

The loop-hafnian, a generalization of both the permanent and hafnian, introduces unique challenges as its corresponding quantum states require Dicke-like superpositions, necessitating non-trivial quantum circuit designs. This unification not only strengthens the theoretical underpinnings of quantum circuit models but also reveals new, diverse applications that are inherently difficult for classical computers to solve.

Quantum computers are anticipated to surpass classical computers, driving interest in demonstrating quantum computational advantage. A key approach involves quantum sampling, specifically sampling from the output distributions of a quantum system believed to be intractable for classical computation. Boson sampling, Gaussian boson sampling, and quantum spin models represent prominent examples of this approach, each with its own strengths and limitations.

While boson sampling faces challenges related to interferometer size and photon sources, Gaussian boson sampling relies on squeezing parameters that can limit performance. Spin model-based sampling offers advantages such as the absence of the hiding property and deterministic initial state preparation.

However, existing spin model research has been restricted to bipartite interactions, limiting its applicability. This study overcomes this limitation by generalizing the interaction structure, allowing for the representation of permanents, hafnians, and loop-haufnians through transition amplitudes. The ability to encode arbitrary real symmetric matrices through interaction strengths within the spin model provides a versatile platform for exploring classically intractable computations.

Ising model construction and Gaussian boson sampling with squeezed states represent promising avenues for quantum computation

Researchers investigated the relationship between bosonic counting problems, spin models, and matrix functions such as the permanent, hafnian, and loop-hafnian. The study began by constructing an Ising model extended to arbitrary interaction networks, demonstrating that transition amplitudes within the Ising Hamiltonian are proportional to the hafnian and loop-hafnian.

These matrix functions are linked to the output probabilities of the system, establishing a connection between quantum computation and classically hard problems. Gaussian boson sampling (GBS) utilising squeezed states, rather than single-particle states, was employed to generate output probabilities proportional to squared hafnians or loop-hafnians of submatrices.

These submatrices are determined by interferometer settings, squeezing parameters, and displacement parameters. Unlike single-photon sources, squeezed states offer deterministic preparation, and GBS has been experimentally demonstrated across platforms including those used for quantum chemistry and graph-related problems.

However, performance is currently limited by factors such as squeezing parameters, loss, and inefficiency. To circumvent these limitations, the work also explored quantum sampling based on spin models, specifically a balanced bipartite Ising model with 2N spin-1/2 systems. Transition amplitudes from the all-spin-down to the all-spin-up state were calculated, revealing proportionality to the permanent of a real N × N matrix.

This approach avoids the need for the ‘hiding property’ required in boson sampling, allowing for a smaller number of spins compared to the modes needed in boson sampling and enabling deterministic initial state preparation. The researchers extended this analysis to broader classes of spin models, establishing average-case hardness under the anticoncentration conjecture.

Notably, the study demonstrated that the loop-hafnian necessitates Dicke-like superpositions, requiring non-trivial circuit design but remaining potentially realisable on quantum computers. A graphical summary was created to illustrate the correspondences between matrix functions, graph structures, bosonic networks, and spin networks, highlighting how the interaction structure of spin models mirrors the corresponding graph classes. The loop-hafnian case introduces an additional complete graph spin lattice associated with self-loops, reflecting a superposition of spin configurations.

Ising model amplitudes, matrix functions and computational complexity in boson sampling are deeply interconnected concepts

Transition amplitudes of the Ising Hamiltonian are proportional to the hafnian and the loop-hafnian, establishing a unified framework linking bosonic networks with spin dynamics and matrix functions. This work extends the Ising model construction to arbitrary interaction networks, revealing a correspondence between matrix functions and quantum spin systems.

The loop-hafnian, which generalizes both the permanent and hafnian, necessitates Dicke-like superpositions for its corresponding quantum circuits, presenting a non-trivial design challenge. Boson sampling considers linear optical networks with input modes prepared in single-photon or vacuum states, yielding output probabilities proportional to the squared matrix permanents of Gaussian submatrices.

Computing the permanent is known to be #P-hard, suggesting that efficient classical sampling of boson sampling is improbable unless the polynomial hierarchy collapses. Gaussian boson sampling utilises squeezed states instead of single photons, resulting in output probabilities proportional to squared hafnians or loop-hafnians of submatrices determined by interferometer, squeezing, and displacement parameters.

The hafnian can reproduce the permanent with specific matrix structures, encompassing the arguments of boson sampling and enabling deterministic state preparation. Experimental demonstrations of Gaussian boson sampling have been achieved across various platforms, facilitating applications in quantum chemistry and graph-related problems.

However, performance is constrained by squeezing parameters, loss, and inefficiency. Quantum sampling based on spin models offers an alternative, with transition amplitudes of bipartite Ising spin models proportional to the permanent of a real matrix. This approach does not require the hiding property inherent in boson sampling, allowing for matrices beyond random Gaussian forms and eliminating the need for a collision-free condition.

Existing spin model results have been limited to bipartite interactions, corresponding to the permanent. This research generalizes the interaction structure, demonstrating that the permanent, hafnian, and loop-hafnian can be represented through transition amplitudes, potentially connecting quantum spin dynamics with computational hardness. Arbitrary real symmetric matrices can be encoded through interaction strengths within the spin model, offering a versatile approach to quantum simulation.

Ising model transition amplitudes and connections to hafnian and loop-hafnian calculations are explored in detail

Researchers have established a unified framework connecting bosonic networks of single photons and Gaussian states with spin dynamics and matrix functions, demonstrating a significant advance in quantum information science. This work extends the established relationship between Ising model interactions and the permanent to encompass arbitrary interaction networks, revealing that transition amplitudes within the Ising Hamiltonian are proportional to both the hafnian and the loop-hafnian.

These matrix functions, traditionally associated with computational complexity, emerge naturally when describing the dynamics of these quantum systems, suggesting potential applications in classically intractable problems. The findings consolidate permanents, hafnians, and loop-hafnians within a single theoretical structure, broadening the scope of circuit models and highlighting diverse applications with inherent classical computational challenges.

Specifically, the loop-hafnian, a generalization of both the permanent and hafnian, requires Dicke-like superpositions, necessitating non-trivial circuit designs for its implementation. While the authors acknowledge the complexity of preparing the required quantum states for loop-hafnian calculations, they detail a potential circuit implementation and suggest future research could focus on optimising this process.

A limitation noted is the current focus on specific Hamiltonian structures and the associated computational demands of simulating loop-hafnian-based states. Future research directions include exploring efficient methods for generating these states and investigating the potential for utilising this framework in developing novel quantum algorithms. This unification of mathematical structures and physical systems offers a promising avenue for exploring the boundaries of quantum computation and its applications to classically hard problems.