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Voxel-based block variational quantum linear solver: a hybrid quantum-classical method for static analysis of solids
Feng Wu, Chen Li, Li Zhu, Yuxiang Yang, Xu Guo
TL;DR
Large static finite-element problems in solid mechanics produce sparse linear systems, while VQLS faces challenges in matrix decomposition, trainability, and measurement cost. Voxel-BVQLS combines structured LCU decomposition, a minimum-potential-energy objective, and block-Hadamard tests; in three noiseless classical simulations it reduced unitary terms and iterations while producing approximate fields.
Problem
Large static finite-element systems are costly to solve, and VQLS faces matrix-decomposition, barren-plateau, and expectation-value measurement challenges in quantum finite-element analysis.
Source: §1 · pp. 1–2Method
Voxel-BVQLS uses voxel-grid structure for stiffness-matrix LCU decomposition, a minimum-potential-energy variational objective, and block-Hadamard tests.
Source: §6 · p. 30Results
Across three examples, voxel-grid LCU uses 18 nonzero unitary terms in each case versus 51, 224, and 228 Pauli terms, and the method reaches reported field errors in fewer iterations than comparison VQLS methods.
Source: §6 · p. 30Takeaways & Limitations
The examples support Voxel-BVQLS as a structured hybrid quantum–classical framework for finite-element analysis on regular voxel grids.
Source: §6 · p. 30All calculations were classical and noiseless; finite-shot sampling and quantum gate and measurement noise were not included, and multiplexor hardware cost remains to be assessed.
Source: §6 · p. 30
Abstract
from arXiv · showhide
In solid mechanics, finite element discretization of large-scale static problems produces large sparse linear systems whose solution requires substantial computation time and memory. The variational quantum linear solver (VQLS) offers a hybrid quantum-classical route, but its use in quantum finite element analysis is limited by the decomposition of nonunitary matrices, barren plateaus, and the measurement cost of expectation values. We propose a voxel-based block variational quantum linear solver (Voxel-BVQLS) that combines structured matrix decomposition, the principle of minimum potential energy, and batched quantum tests. First, we construct an LCU decomposition of the stiffness matrix from the recursive block-banded structure of voxel-grid finite element matrices, with the number of unitary terms bounded independently of the problem size. Second, we optimize the ansatz parameters using a minimum-potential-energy objective in place of a conventional VQLS loss function, thereby mitigating barren plateaus in the studied problems. Third, we introduce a block-Hadamard test whose circuit directly estimates weighted sums of multiple inner products, reducing the number of circuit configurations required per iteration. We assessed the proposed method in noiseless classical simulations using three examples. These examples show that the method reduces both the number of unitary terms in the LCU decomposition and the number of iterations required to converge, while still yielding solutions of finite accuracy. Voxel-BVQLS thus provides a structured hybrid quantum-classical framework for quantum finite element analysis on regular grids.
1. Introduction
Voxel-BVQLS targets the cost and trainability challenges of applying VQLS to large finite-element systems in solid mechanics. It uses voxel structure, a minimum-potential-energy objective, and block-Hadamard tests; three examples show fewer LCU terms and iterations than comparison methods, with finite-accuracy solutions.
1. Introduction
- Large-scale finite-element discretizations produce sparse systems whose solution time, memory use, and parallel communication overhead typically rise substantially with degrees of freedom.At billions of degrees of freedom, the passage says large-scale parallel computing resources are usually required.Source: §1 · p. 1
- Solid-mechanics stiffness matrices vary with problem structure, and static equilibrium systems are generally nonunitary; end-to-end quantum speedup for general solid mechanics has not been established.The introduction identifies matrix encoding and circuit implementation as challenges for quantum finite-element analysis.Source: §1 · p. 2
- The voxel-grid stiffness-matrix LCU term-count bound is independent of problem dimension, using recursive block-banded structure, cyclic permutations, and Euler’s formula.This is the proposed structured alternative to generic matrix decompositions.Source: §1 · p. 2
- A minimum-potential-energy objective has the unique minimizer K−1f over the full displacement space and can mitigate barren plateaus in the studied settings.The claim applies to constrained symmetric positive-definite stiffness matrices.Source: §1 · p. 2
- The block-Hadamard test estimates weighted sums of multiple inner products in one circuit, reducing circuit configurations required per iteration.It is proposed as part of the block variational solver.Source: §1 · pp. 2–3
- In noiseless classical simulations, structured LCU uses fewer unitary terms than the Pauli decomposition across a two-dimensional truss, plane-stress model, and three-dimensional steady-state heat-conduction model.All three examples produce approximate displacement or temperature fields, with one degree of freedom per node in the heat-conduction model.Source: §1 · p. 3
2. Quantum finite element method and VQLS
This section reviews the hybrid finite-element/VQLS workflow and its quantum representations, then identifies encoding, objective-evaluation, and convergence challenges. It introduces the amplitude-encoded trial state, matrix LCU, VQLS ansatz and losses, and Hadamard-test measurements that underpin the later method.
2. Quantum finite element method and VQLS
- Finite-element assembly remains classical, while a hybrid quantum–classical algorithm solves the discrete algebraic system or estimates selected physical quantities.In the workflow described, measurements estimate an objective and a classical optimizer updates ansatz parameters until a stopping criterion is met.Source: §2 · p. 3
2.1 Quantum amplitude encoding and basic quantum gates
- Amplitude encoding represents a normalized vector of length 2^n as an n-qubit state with computational-basis amplitudes.The text applies this representation to displacement and load vectors.Source: §2.1 · p. 3
2.2 LCU decomposition
- Because the constrained stiffness matrix is nonunitary, its circuit representation uses a weighted sum of unitary matrices, called an LCU decomposition.The common Pauli construction can have 4^n candidate strings, while the number of nonzero terms depends on matrix structure.Source: §2.2 · pp. 4–5
2.3 Overview of VQLS
- VQLS prepares a trial solution with a parameterized ansatz, evaluates a loss through quantum measurements, and updates parameters with a classical optimizer.The framework’s three components are the ansatz, loss function, and measurement procedure.Source: §2 · p. 3, §2.3 · p. 5
2.3.1 Ansatz
- The representative ansatz alternates Ry and CNOT gates to encode a displacement vector, while added layers increase expressivity as well as parameter count, depth, and optimization difficulty.The text frames ansatz design as a balance between expressivity and trainability.Source: §2.3.1 · p. 5, Figure 1 · p. 6
2.3.2 Loss functions
- Global, local, and residual losses measure solution quality in different ways, and their global minima are zero when K is invertible and the ansatz can represent the normalized exact solution.Estimating losses and gradients is identified as the main computational expense during optimization.Source: §2.3.2 · p. 6
2.3.3 Hadamard test
- The Hadamard test maps real parts of unitary expectation values or state overlaps to an ancilla measurement probability, allowing loss terms to be estimated.In this work, the final objectives are real, although individual unitary terms may have complex expectation values whose imaginary parts cancel in the weighted sum.Source: §2.3.3 · pp. 7–8
2.4 Three challenges for VQLS
- Directly expanding common losses containing KTK yields O(M^2) ordered inner products, while generic Pauli representations may use many terms and VQLS losses can stall or poorly track displacement error.The section motivates a structured stiffness-matrix representation, an objective using one action of K, and a block-Hadamard test to reduce circuit configurations.Source: §2.4 · p. 8
3. Quantum finite element method on voxel grids
Voxel-grid regularity yields recursive block-banded stiffness matrices that support LCU encodings whose term bound is independent of node count. The section extends this decomposition across dimensions and uses minimum potential energy as a variational objective.
3. Quantum finite element method on voxel grids
- For fixed spatial dimension and degrees of freedom per node, the voxel-grid LCU term bound does not grow with node count.The bound depends on spatial dimension and nodal degrees of freedom, unlike the generic Pauli decomposition described in the section.Source: §3 · p. 8, §3.1.2 · p. 14
3.1 LCU decomposition of stiffness matrices on voxel grids
- The decomposition uses cyclic permutations, SVD of real 2×2 blocks, and Euler’s formula to express nonnegative diagonal matrices as sums of two diagonal unitaries.The construction is derived for real symmetric stiffness matrices on one-, two-, and three-dimensional voxel grids.Source: §3.1 · p. 9, §3.1.2 · p. 14
3.1.1 LCU decomposition for a one-dimensional voxel grid
- One-dimensional grids require at most six unitary terms for d=2 and 14 terms for d=4.The d=2 construction decomposes each of three matrix components into two unitaries; the d=4 construction uses 14 terms.Source: §3.1.1 · pp. 12–13
3.1.2 LCU decomposition for two- and three-dimensional voxel grids
- Two-dimensional grids require at most 18 terms for d=1 or 2 and 42 for d=4; three-dimensional grids require at most 54 and 126, respectively.The higher-dimensional bounds follow from recursively splitting along the grid directions and applying the one-dimensional decomposition.Source: Table 2 · p. 14, §3.1.2 · p. 14
3.2 Solving finite element stiffness equations by minimum potential energy
- The variational method minimizes discrete total potential energy instead of a residual-based VQLS loss; prior studies found normalized energy differences may require less precision for a given displacement-error target.The objective is optimized over the ansatz parameters.Source: §3.2 · p. 15
- When the ansatz Jacobian has full row rank, its gradient norm can indicate solution error, while its smallest singular value reflects local parameter sensitivity.A smaller smallest singular value can accompany weaker sensitivity and make small gradients and slow convergence more likely.Source: §3.2 · p. 16
4. Block-Hadamard quantum test
The block-Hadamard test uses quantum multiplexors to estimate weighted sums of inner products in a single circuit, reducing circuit configurations per iteration. Its extensions batch potential-energy overlaps and derivatives, while the reduction in configurations does not eliminate repeated sampling.
4.1 Block-Hadamard tests for potential energy and its derivatives
- The total-potential-energy objective needs O(M) term-by-term circuit calls, versus O(M²) for the conventional loss-function overlaps described here.The required overlaps are ⟨u|Ki|u⟩ and ⟨u|Ki|uαj⟩.Source: §4.1 · p. 16
- With B = M, one circuit configuration estimates ⟨u|K|uj⟩; the block test combines multiple Ki overlaps rather than testing each separately.A quantum multiplexor selects the Ki terms within a circuit.Source: §4 · p. 16, §4.1 · p. 18
4.2 Block-Hadamard test for batched derivatives
- When B = M and D = Nα, one circuit configuration estimates all potential-energy derivatives, so their configuration count remains constant as a batch.The method encodes the current state and each once-shifted state in a state-index register.Source: §4.2 · pp. 19, 20, 22
- Fewer distinct circuit configurations do not eliminate sampling costs: per-index probability falls as 1/D, and shots depend on index-space size, variance, and target confidence.Each configuration still requires repeated sampling.Source: §4.2 · p. 22
5. Numerical examples
Three proof-of-concept examples—truss, plane-stress, and heat conduction—assess Voxel-BVQLS in classical numerical simulations. Across them, the voxel-grid decomposition uses fewer unitary terms, and the method reaches the reported field approximations in fewer optimization iterations than comparison VQLS methods.
5. Numerical examples
- Voxel-grid LCU used 18 nonzero unitary terms in each example, versus 51, 224, and 228 Pauli terms for truss, plane-stress, and heat conduction.The reductions were about 65% for the truss and about 92% for each of the other two examples.Source: §5.1 · p. 23, Table 3 · p. 23, §5.2 · p. 25
5.1 Truss problem
- With five layers, Voxel-BVQLS reached low truss displacement error after about 110 iterations, versus about 250, 320, and 370 for VQLS-G, VQLS-R, and VQLS-L.At three layers, none of those three comparison methods found a low-error solution within the plotted range.Source: Figure 13 · p. 24, §5.1 · p. 25
- For the truss, the computed configurations for B = 1, 2, and 4 coincide with the finite element reference in the figure.The results indicate no visible nodal-displacement accuracy loss from changing block size in this noiseless example.Source: §5.1 · p. 25, Figure 14 · p. 25
5.2 Plane-stress problem
- At four layers, Voxel-BVQLS reached about 2.15×10^-3 final displacement error; none of the other methods reached 10^-2 in the plotted range.Voxel-BVQLS reached a 1% displacement relative error earlier at each tested depth.Source: §5.2 · p. 27, Figure 17 · p. 27
5.3 Three-dimensional heat-conduction problem
- For heat conduction, the four-layer error fell below 10^-1 after about 120 iterations and stabilized near 4×10^-2; deeper circuits reached lower error levels.Absolute temperature-error color-scale limits declined from about 0.8°C at four layers to 0.15°C at ten and 0.04°C at sixteen layers.Source: §5.3 · p. 28, Figure 21 · p. 29, Figure 22 · p. 30
6. Conclusions
Voxel-BVQLS combines a voxel-grid stiffness-matrix decomposition, a minimum-potential-energy objective, and block-Hadamard tests. The three noiseless examples report fewer unitary terms and optimization iterations than comparison methods, while hardware noise and gate-level costs remain unassessed.
6. Conclusions
- Voxel-BVQLS uses block-banded voxel-grid structure for an LCU term bound independent of mesh-node count, minimum potential energy, and block-Hadamard tests that reduce circuit configurations per iteration.The objective mitigates barren plateaus in the studied settings; the block test estimates weighted sums of multiple inner products.Source: §6 · p. 30
- Across the three examples, voxel-grid LCU used 18 nonzero unitary terms versus 51, 224, and 228 Pauli terms, and the reported field errors were reached in fewer iterations than with comparison VQLS methods.The examples covered truss, plane-stress, and three-dimensional steady-state heat conduction.Source: §6 · p. 30
- All calculations were classical; finite-shot sampling and quantum gate and measurement noise were not included, and multiplexor costs and hardware compatibility need separate assessment.Future work includes gate-level resource estimates, noisy simulation, and improved optimizers for engineering applications.Source: §6 · p. 30
Appendix 2. Quantum circuit for a diagonal phase unitary
The appendix describes a one-hot lookup implementation for a diagonal phase unitary. It outlines parallel ancilla rotations and uncomputation, while noting that implementation costs can grow with the number of distinct phases.
Appendix 2. Quantum circuit for a diagonal phase unitary
- The phase circuit uses a lookup oracle to encode each input’s phase class in a one-hot ancilla register, applies parallel rotations, then uncomputes the register.The phase classes correspond to the distinct values among the input phases.Source: Appendix 2. Quantum circuit for a… · p. 32, Figure 23 · p. 33
- The parallel rotation layer has depth O(1), but requires Ns ancillas and O(Ns) rotation gates; in the worst case, Ns = N.The figure’s space-for-time tradeoff does not establish practical complexity for a particular data-access model or hardware connectivity.Source: Appendix 2. Quantum circuit for a… · p. 33, Figure 23 · p. 33
- A unitary implementation exists, but its actual complexity must be reassessed for the chosen data-access model and hardware connectivity.Without structured g(i) or available quantum random-access memory, generic oracle synthesis generally requires resources growing with N.Source: Appendix 2. Quantum circuit for a… · p. 33
Appendix 3. Block-Hadamard test for the global loss
The global-loss construction groups LCU terms into blocks so a circuit can estimate weighted sums of overlaps. Blocking reduces circuit configurations, with a tradeoff in circuit width, depth, and individual outcome probabilities.
Appendix 3. Block-Hadamard test for the global loss
- Estimating each overlap separately requires O(M^2) circuit calls; partitioning M terms into blocks of B defines a weighted sum for each block pair.Zero coefficients pad the term list when B does not divide M.Source: Appendix 3. Block-Hadamard test for… · p. 33
- Each c_k,l is a weighted sum of B^2 overlaps, and directly estimating it reduces configurations per loss evaluation to O((M/B)^2).Figure 24 shows the B = 2 circuit, with coefficient-state preparation and block SELECT operations in the Hadamard-test branches.Source: Appendix 3. Block-Hadamard test for… · p. 33, Figure 24 · p. 34
- For real symmetric K, summing the block estimates gives the required real quadratic form; B = M uses one configuration but increases circuit width and depth and lowers individual outcome probabilities.The configuration count falls to one when the block contains all M terms.Source: Appendix 3. Block-Hadamard test for… · p. 34