Trading classical and quantum computational resources (1506.01396v1)

Abstract: We propose examples of a hybrid quantum-classical simulation where a classical computer assisted by a small quantum processor can efficiently simulate a larger quantum system. First we consider sparse quantum circuits such that each qubit participates in O(1) two-qubit gates. It is shown that any sparse circuit on n+k qubits can be simulated by sparse circuits on n qubits and a classical processing that takes time $2^{O(k)} poly(n)$. Secondly, we study Pauli-based computation (PBC) where allowed operations are non-destructive eigenvalue measurements of n-qubit Pauli operators. The computation begins by initializing each qubit in the so-called magic state. This model is known to be equivalent to the universal quantum computer. We show that any PBC on n+k qubits can be simulated by PBCs on n qubits and a classical processing that takes time $2^{O(k)} poly(n)$. Finally, we propose a purely classical algorithm that can simulate a PBC on n qubits in a time $2^{c n} poly(n)$ where $c\approx 0.94$. This improves upon the brute-force simulation method which takes time $2^n poly(n)$. Our algorithm exploits the fact that n-fold tensor products of magic states admit a low-rank decomposition into n-qubit stabilizer states.

• The paper demonstrates that hybrid approaches can simulate larger quantum circuits by decomposing operations into manageable classical and quantum components.
• It introduces sparse circuits and non-destructive Pauli measurements to reduce required quantum resources, achieving simulations in 2^(O(k)) poly(n) time.
• The findings imply that efficient quantum simulations are feasible using current hardware, by strategically leveraging classical computation to ease quantum complexity.

Exploring the Tradeoff between Classical and Quantum Resources in Hybrid Simulations

The paper "Trading Classical and Quantum Computational Resources" investigates the potential of hybrid quantum-classical simulations to efficiently simulate larger quantum systems using a combination of a classical computer and a small quantum processor. The authors, Sergey Bravyi, Graeme Smith, and John A. Smolin, introduce two models—sparse quantum circuits and Pauli-based computation (PBC)—and provide methodologies for simulating these models effectively with limited quantum resources by leveraging classical computation.

Sparse Quantum Circuits

The paper begins with an exploration of sparse quantum circuits, where each qubit is involved in a limited number $O(1)$ of two-qubit gates. The researchers establish that any sparse circuit acting on $n + k$ qubits can be decomposed into operations on $n$ qubits with classical processing that runs in $2^{O(k)} \text{poly}(n)$ time. This result is particularly useful when both $k$ and the circuit depth $d$ remain small. The methodology involves decomposing the larger circuit into a linear combination of smaller circuits, thereby distributing computational load between the quantum and classical components.

Pauli-Based Computation

The authors then examine the PBC model, where operations consist of non-destructive eigenvalue measurements of Pauli operators, and initial qubit states are prepared in magic states. The research shows an equivalence between PBC and universal quantum computation. Crucially, they demonstrate that PBCs on $n+k$ qubits can be simulated by PBCs on $n$ qubits with classical computation still in $2^{O(k)} \text{poly}(n)$ time. Moreover, the authors propose a purely classical simulation method for PBCs that runs faster than brute-force approaches, achieving a time complexity of $2^{\alpha n} \text{poly}(n)$ with $\alpha \approx 0.94$.

Implications and Future Directions

This investigation into the tradeoff between classical and quantum resources in hybrid simulations reveals that significant reductions in quantum computational complexity can be achieved with suitable classical algorithms. By reducing the requirement for quantum resources, these methods could make practical quantum simulations feasible sooner, given the current limitations in scalable quantum hardware.

Theoretical implications include an enhanced understanding of the computational power associated with classical simulations of quantum circuits, particularly concerning magic state and stabilizer decomposition. Practically, the techniques presented could guide the development of efficient quantum algorithms, where the computational burden is balanced with classical computation, thereby optimizing available quantum resources.

Future research could explore similar hybrid strategies for other quantum computational models, evaluate the scalability of these methods with increasing qubit numbers, and address the robustness of classical processing under realistic conditions, including noise and decoherence.

Overall, the paper demonstrates that a hybrid quantum-classical approach holds promising potential for advancing the capabilities of quantum simulations and offers novel insights into leveraging classical computational resources in the quantum computing landscape.