Variational Quantum Linear Solver (1909.05820v4)

Abstract: Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare $|x\rangle$ such that $A|x\rangle\propto|b\rangle$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision $\epsilon$ is achieved. Specifically, we prove that $C \geq \epsilon^2 / \kappa^2$, where $C$ is the VQLS cost function and $\kappa$ is the condition number of $A$. We present efficient quantum circuits to estimate $C$, while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of $1024\times1024$. Finally, we numerically solve non-trivial problems of size up to $2^{50}\times2^{50}$. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $\epsilon$, $\kappa$, and the system size $N$.

• The paper introduces a hybrid quantum-classical algorithm that variationally prepares quantum states to solve linear systems on current NISQ devices.
• It demonstrates efficient cost estimation using shallow quantum circuits and validates scalability with hardware experiments up to 1024×1024 systems.
• The study establishes a termination condition linked to matrix condition numbers and precision, ensuring practical quantum advantage for real-world applications.

Variational Quantum Linear Solver: An Efficient Hybrid Algorithm for Near-term Quantum Computers

The paper entitled "Variational Quantum Linear Solver" (VQLS) introduces a promising hybrid quantum-classical algorithm for solving linear systems on near-term quantum devices. Traditional quantum algorithms such as Harrow-Hassidim-Lloyd (HHL), although theoretically capable of exponential speedup, require circuit depths that current noisy intermediate-scale quantum (NISQ) devices cannot achieve. VQLS presents an alternative approach designed to operate within the constraints of these NISQ devices, leveraging the variational techniques that combine shallow quantum circuits with classical optimization methods.

Overview of VQLS

VQLS is structured to variationally prepare a quantum state $|x⟩$ such that $A|x⟩ \propto |b⟩$, where A is the matrix of the linear system, and $|b⟩$ corresponds to the known state vector. The algorithm initiates with a decomposition of A into a linear combination of unitaries and then employs a short-depth variational circuit as an ansatz, parameterized by $#1{\alpha}$ to approximate $|x⟩$. The algorithm iteratively adjusts these parameters using classical optimization techniques until the cost function converges to a specified termination condition ensuring a desired solution precision $\epsilon$.

Key Contributions and Results

1. Operational Termination Condition: The paper derives a meaningful termination condition, proving that the solution precision $\epsilon$ scales with respect to the condition number $\kappa$ of matrix $A$, significantly influencing the precision guarantees of the approach.
2. Efficient Cost Estimation: The paper details efficient quantum circuits to estimate VQLS's cost functions, providing evidence that such estimations are classically hard, thereby justifying the quantum advantage.
3. Practical Implementation: Using Rigetti's quantum computer, the authors implement VQLS successfully for systems up to size $1024\times1024$. Numerical simulations extend these results dramatically, demonstrating system solutions for sizes up to $2^{50}\times2^{50}$, showcasing the scalability of VQLS in terms of quantum resources.
4. Scalability and Performance Analysis: The algorithm shows heuristic evidence of scaling efficiently with the problem's parameters, i.e., linearly with the condition number $\kappa$, logarithmically with the precision $\epsilon$, and polylogarithmically with the system size $N$. This makes it highly suitable for NISQ devices and represents potential for intermediate quantum devices.

Implications and Future Directions

The formulation of VQLS aligns with the current technological trajectory towards practical quantum computing, offering a feasible path to utilizing quantum devices before the advent of fully fault-tolerant quantum computers. This work opens several avenues for future research:

• Enhanced Ansatz Design: While the paper provides a basic ansatz structure, further refinement could improve the expressiveness and performance, employing techniques such as Quantum Alternating Operator Ansatz (QAOA).
• Noise Resilience: The authors claim potential noise resilience of the algorithm, supported by preliminary evidence. A deeper examination into the robustness of VQLS under various NISQ noise models and inclusion of error mitigation techniques could bolster practical applications.
• Algorithmic Extensions: VQLS could be adapted or extended to solve more complex problems beyond linear systems, such as differential equations or matrix inversions inherent in quantum machine learning applications.

This paper represents a significant step in bridging theoretical quantum computing advances with realistic computational applications, particularly in the domain of solving linear algebra problems efficiently on quantum hardware available today. The presented VQLS algorithm is promising for the broader adoption of quantum technologies and serves as a pivotal tool in the era of NISQ computing.