[Paper Review] Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm

Dong An, Lin Lin|arXiv (Cornell University)|Sep 12, 2019

Quantum Computing Algorithms and Architecture44 references85 citations

TL;DR

This paper proposes a time-optimal adiabatic quantum computing (AQC) framework and its variant via quantum approximate optimization algorithm (QAOA) to solve quantum linear systems (QLSP) with near-optimal complexity. By rescheduling the adiabatic path using a tunable function, the method achieves O(𝜅poly(log(𝜅/𝜖))) runtime for general non-Hermitian matrices, eliminating the need for complex amplitude amplification and outperforming vanilla AQC and randomization methods in numerical tests.

ABSTRACT

We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with $\mathcal{O}(\kappa~ ext{poly}(\log(\kappa/\epsilon)))$ runtime, where $\kappa$ is the condition number, and $\epsilon$ is the target accuracy. This is near optimal with respect to both $\kappa$ and $\epsilon$. Our method is applicable to general non-Hermitian matrices, and the cost as well as the number of qubits can be reduced when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) with an optimal control protocol can also achieve the same complexity in terms of the runtime. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method.

Motivation & Objective

• To develop a time-optimal adiabatic quantum computing (AQC) approach for solving the quantum linear system problem (QLSP) with improved runtime complexity.
• To eliminate reliance on complex amplitude amplification techniques such as variable time amplitude amplification (VTAA), which are difficult to implement.
• To demonstrate that QAOA, when combined with optimal control protocols, can achieve the same near-optimal complexity as time-optimal AQC.
• To provide a gate-efficient implementation using truncated Dyson series and standard query models for sparse matrices and state preparation.
• To numerically validate that QAOA achieves the lowest runtime among AQC variants and competing methods.

Proposed method

• Proposes AQC(p) and AQC(exp) scheduling functions that reschedule the adiabatic path to minimize runtime, with p ∈ (1,2) and exponential scheduling for optimal scaling.
• Uses the energy gap information along the adiabatic path to design time-optimal evolution, reducing runtime dependence on condition number 𝜅 and accuracy 𝜖.
• Applies truncated Dyson series for efficient adiabatic dynamics simulation, achieving query complexity O(d𝜅/𝜖 log(d𝜅/𝜖)) for AQC(p) and O(d𝜅 poly log(d𝜅/𝜖)) for AQC(exp).
• Leverages the close relationship between AQC and QAOA to show that QAOA with optimal control can achieve the same runtime complexity as time-optimal AQC.
• Employs standard query models: d-sparse matrix access and state preparation oracle for |b⟩, enabling gate-efficient implementation.
• Numerically evaluates performance across Hermitian positive definite and non-Hermitian matrices, comparing AQC(p), AQC(exp), QAOA, RM, and vanilla AQC.

Experimental results

Research questions

• RQ1 Can adiabatic quantum computing be optimized to achieve near-optimal runtime complexity for QLSP without relying on complex amplitude amplification?
• RQ2 What scheduling function enables AQC to achieve O(𝜅poly(log(𝜅/𝜖))) runtime scaling, and how does it compare to existing methods?
• RQ3 Can the quantum approximate optimization algorithm (QAOA) inherit the same complexity scaling as time-optimal AQC when using an optimal control protocol?
• RQ4 How does the performance of QAOA compare to AQC variants and the randomization method (RM) in terms of runtime and scaling with respect to 𝜅 and 𝜖?
• RQ5 What is the numerical scaling behavior of AQC(p), AQC(exp), and QAOA across different matrix types and parameter regimes?

Key findings

• The AQC(p) method achieves O(𝜅/𝜖) runtime complexity for 1 < p < 2, removing a logarithmic factor in 𝜅 compared to the randomization method (RM), which scales as O(𝜅 log 𝜅 / 𝜖).
• The AQC(exp) method achieves O(𝜅 poly(log(𝜅/𝜖))) runtime, which is near-optimal with respect to both 𝜅 and 𝜖, and does not require prior knowledge of 𝜅.
• Numerical results show that QAOA achieves the lowest runtime among all methods tested, with scaling O(log^1.5(1/𝜖)) in accuracy, outperforming even AQC(exp) for small 𝜖.
• For the condition number 𝜅, QAOA shows a scaling of approximately O(𝜅^0.89), while AQC(exp) scales as O(𝜅^0.93) and AQC(p) as O(𝜅^1.05), indicating near-linear dependence.
• The AQC(exp) method outperforms AQC(p) when 𝜖 is small enough, confirming its poly-logarithmic dependence on 1/𝜖.
• The runtime of QAOA is smaller than AQC(exp) when the optimal parameters are found, suggesting that QAOA dynamics may not strictly follow the adiabatic path, indicating a non-adiabatic advantage.