Structure optimization for parameterized quantum circuits

• Parameterized quantum-circuit methods typically choose a fixed rotation-gate structure and iteratively update its parameters using measured objective values.Source: §1 · p. 1
• Noise generally increases with circuit depth, motivating shallow circuits that remain expressive enough for the task.Source: §1 · p. 1
• Prior structure-optimization approaches used genetic selection of unparameterized gates or iterative growth with gradient-based reoptimization.Source: §1 · p. 1
• The proposed method jointly optimizes circuit structure and objective-function parameters by choosing each gate’s rotation direction and angle.Directions are selected from tensor products of Pauli matrices, while angles remain continuous.Source: §1 · pp. 1–2
• For single-qubit parameterized gates with fixed entangling gates, the method is compatible with existing NISQ hardware and requires seven energy estimations per optimization step.The method extends angle optimization to selecting an optimal direction from canonical generators.Source: §1 · p. 2

Angle optimization

• For one rotation angle, the expectation value is sinusoidal, enabling closed-form coordinate minimization while all other circuit degrees of freedom remain fixed.The method models the objective as ⟨M⟩θd = A sin(θd + B) + C.Source: §2 · p. 2
• Rotosolve sequentially finds each gate’s optimal angle, cycles through all gates, and repeats until a stopping criterion is met.Stopping can be based on a fixed cycle budget or insufficient energy reduction over consecutive cycles.Source: §2 · p. 2

Structure optimization

• Rotoselect relaxes fixed structure by greedily evaluating candidate generators for each gate and retaining the generator-angle pair with the lowest energy.The selected generator and corresponding angle are updated gate by gate across repeated cycles.Source: §2 · p. 3

Hardware constraints

• The general generator search has 4^n choices per gate, but practical implementations restrict candidates to the hardware’s native gate set and often use fixed two-qubit layouts.The paper focuses on single-qubit structure optimization with fixed entangling gates for NISQ compatibility.Source: §2 · p. 3

Structure optimization

• With single-qubit generators restricted to X, Y, and Z, Rotoselect reduces the nominal nine evaluations to seven by sharing the zero-angle identity estimate.The identity generator is unnecessary because zero rotation reproduces its effect.Source: §2 · p. 3

Convergence

• Because measurements make the objective stochastic and multiple generators can attain equally good minima, convergence analysis for Rotoselect is more complicated than for fixed-structure coordinate minimization.Source: §2 · p. 4

3.1 Performance on the variational quantum eigensolver

• Rotoselect achieved better mean, standard deviation, and lowest energy than Rotosolve for all tested Heisenberg circuit depths.The advantage was especially pronounced for lower-depth circuits, while both methods improved as layers increased and remained robust to finite measurement noise.Source: Figure 2 · p. 5, §3.1 · p. 6
• The LiH experiment showed results consistent with the previous experiment, reporting mean and standard deviation of the lowest energy across 10 trials for circuits with up to 7 layers.Using 1000 measurements per Hamiltonian term or exact energy calculations, Figure 3 compared Rotosolve and Rotoselect for VQE optimization of LiH. The 3-layer Rotosolve results were excluded because an outlier put the mean above the plotted range.Source: Figure 3 · p. 5, §3.1 · p. 6

3.2 Demonstration on the IBM Melbourne computer

• On IBM Melbourne, Rotoselect reached close to the Hydrogen ground state within 56 evaluations, although device results were slightly worse on average than simulator results.The experiment used a 2-layer circuit with four adjustable angles, CNOT entanglers, five trials, and 1000 measurements per Hamiltonian term.Source: §3.2 · p. 6, Figure 4 · p. 6

3.3 Optimization of circuits with limited expressibility

• For the unfavorable circuit #15, Rotosolve could not reduce trace distance below a fixed value, whereas Rotoselect approached zero at 7 layers.This experiment optimized overlap with uniformly sampled random target states using exact energy and showed that structure optimization can boost a poor initial circuit structure.Source: §3.3 · p. 7, Figure 5 · p. 7

3.4 Comparison of optimization algorithms and scaling

• Rotosolve and Rotoselect converged with significantly fewer energy evaluations than Adam or SPSA and showed favorable scaling over the tested qubit counts.The comparison used a 5-qubit Heisenberg chain for energy-versus-evaluations results and measured evaluations needed to reach within 2% of the ground-state energy as system size increased.Source: §3.4 · pp. 7–8, Figure 6 · p. 8

4 Discussion

• Rotosolve can jointly optimize K rotation angles by evaluating 3K circuits; although this cost is exponential, the approach remains computationally feasible for small K and may provide an advantage.The approach belongs to coordinate block minimization and was explored for K ∈ {1, 2, 3}, with angle subsets chosen at random.Source: §4 · p. 8
• Rotoselect can grow circuits incrementally, remove redundant gates, and potentially learn entangling-gate connectivity layouts.Connectivity optimization balances expressive low-depth layers against entangling-gate speed and errors; for n qubits, non-directional two-qubit choices number n(n −1)/2 per layer.Source: §4 · p. 8
• Reusing a previously evaluated energy can reduce evaluations from 3 to 2 for Rotosolve and from 7 to 6 for Rotoselect.The shortcut is exact with infinitely many measurements but introduces noise in realistic scenarios, motivating further investigation.Source: §4 · p. 8
• Because Rotoselect usually changes structure early, switching to Rotosolve afterward could further reduce circuit evaluations.The proposed switch depends on detecting when structural changes have subsided.Source: §4 · p. 9
• Structure optimization improves the expressivity of low-depth circuits with only a small computational overhead, supporting near-term quantum-computing use.The stated motivation is that circuit depth and optimization time are significant bottlenecks on near-term hardware.Source: §4 · p. 9

A Sinusoidal form of expectation values

• The expectation of a Hermitian observable as a function of one rotation angle is sinusoidal with amplitude A, phase B, intercept C, and period 2π.The construction uses Hermitian, unitary generators such as tensor products of Pauli matrices and absorbs surrounding gates into the state and measurement operator.Source: A Sinusoidal form of expectation values · pp. 10–11
• The phase estimator must use arctan2 because commuting cases can make the standard arctangent’s denominator zero.The passage specifically notes that when H commutes with M or ρ, the relevant expression evaluates to zero.Source: A Sinusoidal form of expectation values · p. 11
• Figure 7 illustrates the sinusoidal expectation and shows both expectation and variance minimized near θ∗≈−0.5 for the second angle.The figure also attributes the large variance in panel (a) to that angle’s inability to produce an eigenstate of ZXZX.Source: Figure 7 · p. 12
• The minimizer is expressed in closed form, with k chosen in practice so that θ∗∈(−π, π].Source: A Sinusoidal form of expectation values · p. 12
• Three circuit evaluations suffice to estimate the sinusoid’s coefficients and characterize its minimizer and minimum energy in closed form.The resulting minimum is ⟨M⟩θ∗= −A+C and can be extrapolated without additional cost.Source: A Sinusoidal form of expectation values · p. 12

Scope and limitation

• The generalized generator construction extends the gate family beyond the canonical Pauli-matrix rotation axes used by the basic algorithms.The derivation is general, while the practical algorithms rely on canonical Pauli axes.Source: B A few generalizations · p. 13

Single-qubit gates

• Arbitrary single-qubit axes can be used by choosing H_d = c_xX + c_yY + c_zZ with (c_x, c_y, c_z) a unit vector.This preserves the condition needed for sinusoidal expectation values.Source: B A few generalizations · p. 13
• Randomly sampling the coefficient unit vector yields a Rotosolve variant that can search for useful rotation axes, though compiling U_d may require lower-level gates on NISQ hardware.The angle θ_d is still optimized with Algorithm 1.Source: B A few generalizations · p. 13

Multi-qubit gates

• For arbitrary-qubit gates, any unitary V conjugating a tensor product of n Pauli matrices provides an alternative valid generator.The construction starts from H_d ∈ {I, X, Y, Z}^⊗n.Source: B A few generalizations · p. 13