Simulating Large Quantum Circuits on a Small Quantum Computer (1904.00102v2)

Abstract: Limited quantum memory is one of the most important constraints for near-term quantum devices. Understanding whether a small quantum computer can simulate a larger quantum system, or execute an algorithm requiring more qubits than available, is both of theoretical and practical importance. In this Letter, we introduce cluster parameters $K$ and $d$ of a quantum circuit. The tensor network of such a circuit can be decomposed into clusters of size at most $d$ with at most $K$ qubits of inter-cluster quantum communication. We propose a cluster simulation scheme that can simulate any $(K,d)$-clustered quantum circuit on a $d$-qubit machine in time roughly $2^{O(K)}$, with further speedups possible when taking more fine-grained circuit structure into account. We show how our scheme can be used to simulate clustered quantum systems -- such as large molecules -- that can be partitioned into multiple significantly smaller clusters with weak interactions among them. By using a suitable clustered ansatz, we also experimentally demonstrate that a quantum variational eigensolver can still achieve the desired performance for estimating the energy of the BeH$_2$ molecule while running on a physical quantum device with half the number of required qubits.

• The paper presents a cluster-based simulation scheme that partitions circuits into smaller clusters to enable efficient simulation on devices with limited qubits.
• It employs tensor network decomposition and an edge-cutting procedure to replace quantum communication with classical computations, thereby optimizing simulation efficiency.
• Applications to Hamiltonian simulation and VQE demonstrate the method's practical impact in estimating molecular energies and reducing circuit depth.

Simulating Large Quantum Circuits on a Small Quantum Computer

The paper "Simulating Large Quantum Circuits on a Small Quantum Computer" presents a novel approach to efficiently simulate large quantum circuits on quantum computers with limited qubit availability, addressing a critical constraint in near-term quantum computing technologies. The authors introduce the concept of cluster parameters, denoted as $K$ and $d$, which enable the decomposition of a quantum circuit's tensor network into multiple smaller clusters, facilitating simulation on quantum hardware with $d$ qubits.

Overview of Clustered Quantum Circuits and Tensor Networks

In scenarios where quantum memory is limited, the ability to simulate quantum systems that exceed the physical qubit capacity is crucial. Through tensor network decomposition, the authors propose a cluster-based simulation scheme allowing circuits to be partitioned into clusters, each with a size at most $d$ and inter-cluster communication restricted to $K$ qubits. This framework leverages the mathematical structure of tensor networks, a representation of quantum circuits in terms of graphs and tensors, to outline the clustering process effectively.

To relate the mathematical formulation to practical performance, circuits are described as either $(K,d)$-clustered or having specific interaction graphs. The main challenge is ensuring that the simulation does not compromise operations across clusters. By decomposing the tensor network using an edge-cutting procedure, quantum communication is replaced with classical computations, trading off some computational efficiency for reduced quantum circuit scale.

Simulation Efficiency and Complexity Analysis

The authors have characterized the complexity of simulating clustered quantum circuits, providing an efficiency analysis showing that their method enables simulations in time $2^{O(K)}$ coupled with classical processing. For decomposable functions, optimized by contraction complexity $\cc(g)$, the simulation further improves. The authors assert that a clustered circuit represented by the topology $(K,d)$ achieves simulation capabilities on a $d$-qubit machine, with polynomial scaling in the number of qubits and gates, and introduce optimization techniques depending upon circuit-specific parameters.

Significantly, simulations considering specific post-processing functions and clustered quantum systems featuring a graph topology featuring limited inter-cluster interactions make the approach viable for systems such as large molecules, whose quantum characteristics can be broken down into smaller interacting parts.

Applications to Hamiltonian Simulation and VQE

Leveraging the introduced simulation scheme, the paper applies its methodology to Hamiltonian simulations of clustered quantum systems and Variational Quantum Eigensolver (VQE) techniques. Hamiltonian simulation, critical for harnessing quantum advantages in material science, is demonstrated with examples where interaction strength $h$ and evolution time $t$ dictate simulation complexity. These simulations provide theoretically motivated results, estimating correlation functions within bounded error limits.

The authors also explore VQE as an interface between classical and quantum processing, showcasing experiments on small-scale quantum devices. With reduced qubit requirements, the VQE experiments estimate molecular ground energies like those of the BeH$_2$ molecule. The reduced-depth circuits validate the computational efficiency gains from simulating only parts of a full quantum circuit at a time.

Implications and Future Directions

With quantum computing poised at the cusp of practical application in industry, the presented framework represents a strategic pathway to maximize existing hardware limitations, while paving avenues for research in modular quantum system design and hybrid algorithms in quantum optimization. For systems capable of weak inter-cluster interactions, the proposed strategies offer new opportunities to decompose large computational problems, fundamentally shifting the computational load between quantum and classical domains.

Future developments will likely focus on optimizing the selection of cluster parameters based on problem-specific tensor network styles, addressing graph partitioning complexities, and realizing simulation strategies capable of broader quantum algorithm applications. Additionally, experimental validation and comparisons with other qubit-efficient schemes will be crucial in setting benchmarks for hybrid quantum-classical simulator performance.

Conclusion

In summary, the paper tackles a prevailing problem in quantum computation—simulating circuits beyond current qubit limitations—with an innovative clustering scheme. This not only emphasizes the strategic use of tensor networks but also highlights practical implementations in quantum simulation and optimization. Its implications stretch across several domains, promising improvements in quantum applications despite hardware constraints.