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Effects of Trotter Error, Digitization Error, and Initial State Overlap on Tapered Quantum Phase Estimation for Minimum Eigenvalue Computation
Elijah Pelofske, Stephan Eidenbenz
TL;DR
Tapered QPE’s practical error behavior in minimum-eigenvalue computation depends on implementation and input conditions. The paper numerically implements full trotterized circuits and finds that overlap sets steady-state optimal-phase sampling, while Trotter error can cause non-physical estimates.
Problem
The paper examines how practical factors affect tapered QPE error rates when computing a quantum Hamiltonian’s minimum eigenvalue.
Source: §I · p. 1Method
The authors numerically simulate full trotterized tapered-QPE circuits using an approximate optimal taper and examine representative small Hamiltonians.
Source: §I · p. 1, §II · p. 3Results
Steady-state optimal-phase probability is determined by ground-state overlap; low-Trotter-error tQPE has no strong evolution-time dependence, while Trotter error can yield non-physical eigenvalue estimates.
Source: §III · p. 6, §IV · pp. 11–12Takeaways & Limitations
Trotterization need not be arbitrarily expensive to reach overlap-predicted steady-state sampling, but poor unitary approximations can produce non-physical estimates.
Source: §III · pp. 6–7The reported circuit executions focus on single 4-qubit and 3-qubit instances of the Heisenberg quantum Hamiltonian model.
Source: §II · p. 3
Abstract
from arXiv · showhide
We numerically implement full quantum circuits of optimal tapered Quantum Phase Estimation (in the form of an implementable approximate bandwidth-limited DPSS taper) for the purpose of minimum eigenvalue computation, where the Hamiltonian time-evolution unitary is implemented using standard Trotterization, and the taper state is constructed using an optimized bandwidth-limited taper. We illustrate, on representative quantum Hamiltonians, with tractable exact classical numerical quantum circuit simulations, how tapered QPE performs with respect to optimal phase sampling rate, absolute error, Shannon entropy of the optimal phase distribution, when the tQPE algorithm parameters are changed. Those algorithm parameters are Trotter error, finite sampling, imperfect initial states, total evolution time, and digitization error (number of phase qubits). The reported numerical experiments include up to $\ell=10$ phase qubits of precision with $m=3$ additional phase-register error suppression qubits, on a $4$-qubit Heisenberg quantum magnet model Hamiltonian, using up to 12th-order Trotterization. We numerically show i) the steady-state optimal phase sampling rate is determined by the initial state overlap with the ground-state, ii) with respect to phase sampling probability, or overall eigenvalue sampling error rate, there is no strong evolution time dependence for tQPE unless there is high Trotter error, iii) the approximated time-evolution unitaries in tQPE, like in standard QPE, can result in substantial ringing, which leads to non-physical eigenvalue estimates.
I. INTRODUCTION
This study numerically examines how implementation and input factors affect tapered QPE for minimum-eigenvalue computation. It focuses on ground-state overlap, Trotter error, and phase-register digitization, while noting that standard QPE also depends on evolution time.
I. INTRODUCTION
- QPE estimates minimum eigenvalues of quantum Hamiltonians, and optimal tapered QPE uses a DPSS state to suppress ringing and non-physical errors.The introduction also identifies applications in quantum chemistry, strongly correlated materials, condensed matter physics, HHL, and Shor’s algorithm.Source: §I · p. 1
- The study numerically implements end-to-end trotterized tapered-QPE circuits on small Hamiltonians to examine practical error mechanisms.The simulations emulate large-scale quantum-computer execution for ground-state minimum-eigenvalue computation.Source: §I · p. 1
- The reported tQPE error rate depends on ground-state overlap, Trotter error in controlled-unitary approximations, and phase-register digitization.Digitization error is determined by the number of phase-register precision bits.Source: §I · p. 1
- Unlike standard QPE, tQPE’s three listed error factors exclude evolution time, which the introduction identifies as a fourth factor for standard QPE.The study presents numerical executions to explain how the three tQPE factors influence minimum-eigenvalue error.Source: §I · p. 1
II. METHODS
The method implements tapered QPE with trotterized Hamiltonian evolution and an approximate bandwidth-limited taper state, then examines how phase precision, error suppression, and execution choices affect minimum-eigenvalue estimates. The numerical study uses small Heisenberg-model instances and evaluates phase sampling, eigenvalue error, and phase-distribution entropy.
II. METHODS
- Controlled Hamiltonian evolution is approximated with standard Trotterization, using higher order p and more steps r to reduce simulation error at increased circuit cost.The study implements circuits for small systems that can be simulated with classical statevectors.Source: §II · p. 2, Figure 1 · p. 2
- The simulations use single 4-qubit and 3-qubit disordered Heisenberg-model instances, with independently randomized ±1 couplings.The small system sizes make exact classical diagonalization and circuit statevector simulation tractable.Source: §II · pp. 2–3
- The ℓ phase qubits set digitization precision, while m additional phase qubits suppress ringing; together they provide ℓ + m measured bits.For the experiments, m = 3 and ℓ ranges from 3 to 10.Source: §II · p. 3
- The phase register uses an approximate bandwidth-limited DPSS taper state, chosen for efficient preparation and near-optimal performance relative to the exact DPSS taper.The exact taper could be costly to implement.Source: §II · p. 3
- The base evolution time is selected to avoid phase aliasing, while the study varies it and measures optimal-phase sampling, eigenvalue absolute error, and Shannon entropy.Entropy characterizes whether probability among the optimal phases is concentrated or spread out.Source: §II · pp. 5–6
- Initial-state overlap is measured against the ground-state subspace, which can include degenerate minimum-eigenvalue eigenvectors.Degenerate eigenvalues are identified using a precision of 1 × 10^-12.Source: §II · p. 6
- Measured phase bitstrings are normalized, shifted, and decoded into eigenvalue estimates using the base evolution time.Decoded estimates can be non-physical when approximate trotterized unitaries produce ringing.Source: §II · pp. 4, 6
III. RESULTS
The results show that Trotter error can produce non-physical eigenvalue estimates, while precision, initial-state overlap, and evolution time shape tQPE sampling and error. At sufficiently low Trotter error, optimal-phase sampling approaches the ground-state overlap, though higher phase precision requires more Trotter resources.
Trotter error and eigenvalue estimates
- High Trotter error can produce eigenvalue estimates outside the target Hamiltonian’s spectral range because the implemented unitary is approximate, with intermediate errors also producing side-lobes.tQPE suppresses leakage from phase-grid misalignment, but Trotter error introduces a distinct leakage mechanism.Source: §III · p. 6
- With a perfect initial state, sufficiently low Trotter error drives Φ* sampling to 1, but absolute error reaches a floor from digitization and finite sampling; many large-p runs were incomplete due to memory limits.The circuits could be too large to define or simulate on a single compute node.Source: §III · p. 7
- At second-order Trotterization with r=200, ℓ=3 yields 10^-3–10^-2 error, while ℓ=5 is just below 10^-1; greater precision requires more Trotter resources.The authors identify this as a digitization-error versus Trotter-error trade-off for practical tQPE implementations.Source: §III · pp. 7–8
Evolution-time dependence
- For a perfect initial state, sufficiently low Trotter error brings Φ* sampling to 1 with little evolution-time dependence; high error can produce local trends and periodic sampling changes.The set of optimal phases can change with evolution time, so the reported Φ* probability need not track the same phase throughout.Source: §III · p. 8, Figure 7 · p. 8
- At steady state, the optimal-phase distribution has Shannon entropy ≈0.58 and concentrates near the physical minimum-eigenvalue phase; high Trotter error can make entropy vary from 0.4 to 1.The steady-state distribution is neither concentrated on one phase nor uniform; probability drops for other nearby phases.Source: §III · pp. 8, 10, Figure 8 · p. 9
Initial-state overlap
- With a graph-state initial state of overlap χ≈0.708, Φ* sampling converges to χ at low Trotter error when ℓ resolves the ground and first-excited states; convergence need not be monotonic.Larger ℓ requires more Trotter resources, and the steady state is approximate because of finite sampling, residual Trotter error, and taper-state spectral leakage.Source: §III · pp. 7, 10, Figure 9 · p. 10
IV. CONCLUSION
The simulations find that tQPE’s steady-state optimal phase sampling probability is determined by initial-state overlap, while Trotter error can alter sampling behavior and its dependence on evolution time. The conclusions are bounded by small, single-instance Heisenberg-model simulations and omit phase randomization.
IV. CONCLUSION
- Even with m = 3, small ℓ can produce relatively large digitization error because the high-confidence phase region contains seven digitized phases.This concerns the minimum-eigenvalue estimate E0.Source: §IV · p. 11
- The numerical study targets minimum-eigenvalue computation, a stringent QPE task; sampling low-energy eigenvalues rather than strictly the minimum may suffice in general.The paper notes the minimum-eigenvalue task is application-relevant in computational physics and chemistry.Source: §IV · p. 11
- Steady-state optimal phase sampling probability is determined by initial-state overlap, and Trotter error need not be arbitrarily small to reach that rate.The paper identifies quantifying when steady-state sampling occurs as future work.Source: §IV · p. 11
- The study covers only two Heisenberg spin-glass models with at most four qubits, using single instances to isolate tQPE error sources.The small size is necessitated by the circuit simulations’ computational demands.Source: §IV · p. 11
- Larger systems raise circuit-complexity costs for state preparation and controlled time evolution, and may require higher ℓ to distinguish low-energy eigenstates.The authors focus here on phase-sampling characteristics in small instances; phase randomization was not implemented.Source: §IV · p. 12
- With low Trotter error, tQPE has no strong evolution-time dependence; with Trotter error, dependence appears without a clear periodicity or trend direction.The overall effects of Trotter error on sampling error, phase distribution, and time dependence are not clearly predictable.Source: §IV · p. 12
Appendix A: Taper State Numerical Calculation Details
Appendix A describes the numerical construction of the approximate taper state used to initialize the tQPE circuit. Its bandwidth-limited DPSS optimization targets a prescribed number of optimal phases and is computationally bottlenecked by exact diagonalization.
Appendix A: Taper State Numerical Calculation Details
- The tPREP taper vector is obtained with a numerical heuristic for optimized bandwidth-limited DPSS taper-state preparation.The appendix introduces numerical vector calculations for ℓ = 3 and ℓ = 10.Source: Appendix A · p. 12
- The optimization targets 2m − 1 optimal phases and creates a vector of that size, padding a zero entry in its middle for circuit initialization.ℓ does not change the vector’s size, though its entries change slightly with ℓ.Source: Appendix A · p. 12
- The heuristic is bottlenecked by exact diagonalization, whose matrix size scales with m; the authors report that m ≤ 5 provides sufficient error suppression.Increasing ℓ also increases computation time in practice.Source: Appendix A · p. 12
Appendix B: Heisenberg Model Values
Appendix B supplies the exact weights for the arbitrarily chosen quantum Hamiltonians used in the numerical study, including coupling terms for the model instances.
Appendix B: Heisenberg Model Values
- The appendix provides exact model weights for the arbitrarily chosen Hamiltonians used in the numerical calculations.The supplied passages do not specify the complete weight lists.Source: Appendix B · p. 12
- For the 3-qubit model instance, the supplied passage identifies Jx among the listed weights but does not specify which sites the coupling connects.Source: Appendix B · p. 12