Classical combinations of quantum states for solving banded circulant linear systems

TL;DR

This work addresses solving linear systems where the coefficient matrix is a $K$-banded circulant, a structure common in physics-inspired PDE discretizations. It adapts the classical combinations of quantum states (CQS) approach to express the solution as a classical combination of a polynomially many quantum states, with a convergence guarantee governed by $T = O(K \kappa_C \log(\kappa_C/\nu))$ and a convex optimization step to determine the combination coefficients. The authors provide both a quantum-hybrid circuit implementation and a quantum-inspired classical variant, along with explicit measurement bounds and a detailed procedure to obtain the coefficients that approximate the least-squares solution. Numerical experiments on a one-dimensional heat transfer PDE demonstrate accurate approximations and feasibility on near-term quantum hardware, illustrating a practical route toward early fault-tolerant quantum linear solvers for physics problems.

Abstract

Solving linear systems is of great importance in numerous fields. Proposed quantum algorithms for preparing solutions for linear systems include the HHL algorithm with subsequent refinements and variational methods. Circulant linear systems appear in many physics-related differential equations. An interesting case is banded circulant linear systems whose non-zero terms are within distance K of the main diagonal. For these systems, we propose an approach based on the classical combination of quantum states (CQS) method relying on convex optimization against the available analytical solution. From decompositions into cyclic permutations, the solution can be approximately represented by a classical combination of a polynomial number of quantum states. We validate our methods using classical simulations as well as execution on an IBM quantum computer. While in the setting of this paper, efficient classical algorithms are available, our results demonstrate the potential applicability of the CQS method for solving physics problems such as heat transfer.

Figures (9)

• Figure 1: Here we provide an illustrated overview of our method to solve $K$-banded circulant linear systems. Note the fact that $C$ can be written as a matrix polynomial in terms of the cyclic permutation matrix $Q$ given in \ref{['eqPermute']} with the powers being bounded between $-K$ and $K$. In step (1), we set a truncation threshold $T\ge K$ to create a matrix polynomial with powers of $Q$ from $-T$ to $T$ that serves as an approximation to $C^{-1}$. To find the optimal coefficients of the matrix polynomial $\alpha$, in step (2), we calculate the overlaps between individual components $CQ^m\bm b$ as well as the overlap between $CQ^m\bm b$ and $\bm b$. Such overlaps can be computed with either the Hadamard test for the hybrid quantum-classical algorithm or sample and query access for the quantum-inspired algorithm. Lastly, in step (3), we perform classical convex optimization to find the optimal coefficients $\alpha$.
• Figure 2: Circuit implementation of quantum subroutines in our algorithm. The cyclic permutation matrix $Q$ can be diagonalized by the DFT matrix $F$ such that $Q = F^{-1}\Lambda F$. DFT can be implemented by QFT circuits. The diagonal matrix $\Lambda$ can be implemented as a tensor product of phase gates as shown in \ref{['figIncrementor']}. Powers of $\Lambda$ can be implemented with the same phase gates, by multiplying rotation angles as shown in \ref{['figIncrementorPower']}. To retrieve the real and imaginary parts of $\braket{b|Q^m|b}$, we use the Hadamard test (on the state $\mathsf{QFT}\ket{b}$), which requires a controlled version of $\Lambda^m$, i.e. controlled phase gates, as shown in \ref{['figCompleteRoutineHadamard']}.
• Figure 3: State preparation circuit of $\ket{b}$ modeled after the QAOA circuit farhi2014quantumhadfield2019quantum.
• Figure 4: Numerical results of the algorithm proposed in our paper for solving heat conduction linear systems. First, we show that simulation results of our hybrid and quantum-inspired algorithms can provide decent approximations to the results obtained by matrix multiplication in \ref{['figSimulationresults']}, with errors noticeable only in the logarithmic scale. Further, our hybrid algorithm is executable on IBM quantum hardware as shown in \ref{['figQuantumresult']}. Lastly, \ref{['figCondition']} shows that truncation thresholds fall within our provided upper bound in \ref{['propGuarantee']} in practice.
• Figure 5: Circuit implementation of the Hadamard test. \ref{['figRealHadamard']} is the circuit for retrieving $\mathop{\mathrm{Re}}\nolimits\braket{\psi|U|\psi}$, while \ref{['figImagHadamard']} is the circuit for retrieving $\mathop{\mathrm{Im}}\nolimits\braket{\psi|U|\psi}$.

Theorems & Definitions (14)

• Definition 1: Circulant matrix
• Definition 2: $K$-banded circulant matrix chen1987solution
• Proposition 3
• Proposition 5: Number of measurements needed
• proof
• proof : Proof for \ref{['propGuarantee']}
• Lemma A.1: Matrix Bernstein inequality; Corollary 6.1.2 tropp2015introduction
• Lemma A.2: Vector Bernstein inequality; Lemma 18 kohler2017subsampled
• Proposition A.3: Matrix norm bound
• proof