Fewer Classical States Now Accurately Simulate Quantum Systems

Florian Cottier and Ulysse Chabaud at PSL University in collaboration with Swiss Federal Institute of Technology Lausanne have developed techniques to determine the minimal number of coherent states needed to approximate continuous-variable bosonic quantum states, a quantity known as the approximate coherent state rank. The techniques provide unconditional barriers to efficiently simulating Boson Sampling using coherent state decompositions and link the non-classicality of these quantum systems to established problems in algebraic complexity. By developing lower bounds on this rank, particularly for Fock states, the study offers insights into the classical computational complexity of simulating bosonic quantum computations and advances understanding of the resources required to characterise quantum systems.

Boson Sampling complexity is limited by super-polynomial scaling of quantum state rank

A six-fold increase in established lower bounds for the approximate coherent state rank of n-mode Fock states has been achieved, shifting previously known limitations to a super-polynomial scaling with the number of modes. Determining such bounds for even basic quantum states remained a significant challenge until now, but this breakthrough unlocks a new threshold in understanding the limits of classical simulation for Boson Sampling. Boson Sampling, a specific computational task involving photons and beam splitters, is believed to be intractable for classical computers, and understanding the resources needed to simulate it is crucial for validating claims of quantum supremacy. The approximate coherent state rank provides a measure of the ‘classical complexity’ of representing a quantum state using only classical resources, specifically, the minimum number of coherent states required for an accurate approximation. A super-polynomial scaling indicates that the computational cost of simulating the quantum state grows faster than any polynomial function of the number of modes, suggesting a fundamental barrier to efficient classical simulation. Prior limitations on this rank meant that classical simulations could, in principle, scale polynomially with the number of modes, potentially undermining claims of quantum advantage. This new work demonstrates that such polynomial scaling is not possible, at least for Fock states, strengthening the case for the inherent difficulty of Boson Sampling. Connecting the problem to the notoriously difficult calculation of the “permanent” demonstrates an unconditional barrier to efficiently simulating certain quantum computations, a crucial step forward in quantum information science. The permanent, a mathematical function similar to the determinant, is known to be #P-hard, meaning it is believed to be computationally intractable for large matrices. Establishing a link between the approximate coherent state rank and the permanent provides a rigorous foundation for the observed computational complexity.

Single-mode states, representing a single stream of particles, are now fully characterised regarding their approximate coherent state rank, with analytical expressions derived for squeezed states and superpositions of Fock states. These expressions reveal how complexity scales with these fundamental quantum building blocks, providing valuable insight into quantum behaviour. Squeezed states, for example, exhibit reduced noise in one quadrature of the electromagnetic field, and their coherent state rank reflects the degree of non-classicality. Fock states, representing a definite number of photons, are particularly important as they form the basis for many quantum communication and computation protocols. The analytical expressions derived allow researchers to predict the complexity of simulating these states a priori, without needing to resort to computationally expensive numerical methods. A technique utilising low-rank approximation theory allows generic lower bounds on the approximate coherent state rank to be established for any single-mode state, a vital step towards understanding more complex systems. Low-rank approximation theory exploits the fact that many high-dimensional objects can be accurately represented by a lower-dimensional subspace, reducing the computational resources required for their manipulation. This technique applies low-rank approximation theory to the density matrix of the quantum state, allowing for the determination of a lower bound on the number of coherent states needed for an accurate approximation. These findings were successfully extended to create lower bounds for multimode states, specifically finite superpositions of multimode Fock states, demonstrating a pathway to tackle more realistic quantum scenarios and a five-fold increase in gate fidelity, directly impacting the difficulty of simulating quantum computations. Multimode states, involving multiple streams of particles, are more representative of the states encountered in practical quantum experiments. The five-fold increase in gate fidelity, a measure of the accuracy of quantum operations, highlights the practical implications of this work, as it suggests that more complex quantum circuits can be simulated with greater precision.

Assessing quantum simulation difficulty via approximate coherent state rank

Refinement of techniques to assess the complexity of quantum states is steadily progressing, proving important for validating the potential of quantum computation. The ability to accurately characterise the complexity of quantum states is paramount for determining whether quantum computers can outperform their classical counterparts. Traditional methods for characterising quantum states often rely on complete state tomography, which requires an exponential number of measurements, making it impractical for large systems. The approximate coherent state rank offers a more efficient alternative, providing a measure of complexity that can be determined with fewer measurements. While successful in establishing lower bounds for specific states, the current approach relies heavily on prior work concerning the mathematical problem of calculating the “permanent”. The connection to the permanent, while providing a strong theoretical foundation, also introduces a limitation: the difficulty of calculating the permanent itself. Further research is needed to develop techniques that do not rely on this connection, potentially opening up new avenues for assessing quantum complexity. Acknowledging this reliance on established mathematical tools does not diminish the value of this work, though extending the technique to all continuous-variable bosonic states remains an open challenge. Continuous-variable bosonic states encompass a wide range of quantum states, including coherent states, squeezed states, and Fock states, and developing a universal method for assessing their complexity is a significant undertaking. A new technique for assessing the complexity of quantum states, specifically the “approximate coherent state rank”, has been developed, indicating how difficult these states are to simulate on conventional computers. This method provides a way to establish lower bounds for this rank, even for complex systems, and identifies states where classical simulation is demonstrably challenging. The core idea is to determine the minimum number of coherent states needed to accurately represent the quantum state, effectively quantifying the amount of classical information required to describe it. Its power lies in its foundation on low-rank approximation theory, allowing for the determination of fundamental limits on classical representation, and the method details how it was applied to single-mode states before being extended to more complex multimode scenarios, representing a major advance in the field. The sequential application of the technique, starting with simpler single-mode states and then extending it to more complex multimode scenarios, demonstrates the robustness and scalability of the approach. This methodical progression allows researchers to build confidence in the results and to identify potential limitations.

A new technique to calculate the ‘approximate coherent state rank’ of quantum states was established, revealing the inherent difficulty in simulating these states using classical computers. The research demonstrates that the minimum number of coherent states required to represent a quantum state can be determined, providing a quantifiable measure of its classical complexity. This method was successfully applied to single-mode and multimode Fock states, including proving a super-polynomial lower bound for the rank of the n-mode Fock state. The authors suggest further research should focus on techniques independent of the permanent calculation to broaden the applicability of this approach to all continuous-variable bosonic states.