Source-linked AI summary
Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
David Poulin, Angie Qarry, R. D. Somma, Frank Verstraete
TL;DR
How much does truncating high-frequency variation in a time-dependent Hamiltonian change the evolution it generates? They use convolution to implement a low-pass filter, compare evolutions generated by H(t) and ˜H(t), and bound their difference using G’s normalization and symmetry. With σ much smaller than T ∥H∥2, the norm difference between exact evolution and evolution after truncating high-frequency components is negligible.
Problem
How much does truncating high-frequency variation in a time-dependent Hamiltonian change the evolution it generates?
Source: Appendix B · pp. 5–6Method
They use convolution to implement a low-pass filter, compare evolutions generated by H(t) and ˜H(t), and bound their difference using G’s normalization and symmetry.
Source: Appendix B · p. 5Results
With σ much smaller than T ∥H∥2, the norm difference between exact evolution and evolution after truncating high-frequency components is negligible.
Source: Appendix B · p. 6Takeaways & Limitations
The example and general bound indicate that high-frequency terms affect overall evolution only at second order in perturbation theory.
Source: Appendix B · p. 6The integrals are assumed to exist; piecewise continuity suffices, and short intervals of total duration ǫ may be omitted at error at most ǫ∥H∥.
Source: Appendix B · p. 5
Abstract
from arXiv · showhide
We consider the manifold of all quantum many-body states that can be generated by arbitrary time-dependent local Hamiltonians in a time that scales polynomially in the system size, and show that it occupies an exponentially small volume in Hilbert space. This implies that the overwhelming majority of states in Hilbert space are not physical as they can only be produced after an exponentially long time. We establish this fact by making use of a time-dependent generalization of the Suzuki-Trotter expansion, followed by a counting argument. This also demonstrates that a computational model based on arbitrarily rapidly changing Hamiltonians is no more powerful than the standard quantum circuit model.
Appendix A: Average Hamiltonian
Appendix A compares evolution under a time-averaged Hamiltonian with the exact time-ordered evolution and develops a norm bound on their difference.
Appendix A: Average Hamiltonian
- The proof seeks to show that evolution generated by the average Hamiltonian over a time interval is close to the exact time-ordered evolution.The appendix denotes these operators by U Av(∆t) and U(∆t), respectively.Source: Appendix A · p. 5
- The argument tracks the operator-norm difference through X(s) = 1l−U †(s)U Av(s), which starts at zero.Schrödinger’s equation is used to obtain a bound on ∥X(∆t)∥.Source: Appendix A · p. 5
- Writing the Hamiltonian deviation as ∆H(s) = H(s) −HAv lets the bound use the average-Hamiltonian definition and the triangle inequality.The proof changes the order of integration and also invokes Schrödinger’s equation.Source: Appendix A · p. 5
Appendix B: Decoupling principle
Appendix B smooths local Hamiltonian terms with a Gaussian low-pass filter, then bounds the resulting evolution error and examines high-frequency effects.
Appendix B: Decoupling principle
- Convolving each local term with a normal kernel defines a smoothed Hamiltonian whose frequency cutoff is of order σ−1.The terms remain k-body, so the integrals can be carried out classically; piecewise continuity suffices for their existence.Source: Appendix B · p. 5
- The error bound compares the evolution operators generated by H(t) and its smoothed version, using integral bounds and operator-norm estimates.The derivation uses Schrödinger’s equation and ∥H∥ = supt ∥H(t)∥, with H(t) supported on [0, T].Source: Appendix B · p. 5
- If the required integrals fail to exist only on short intervals of total duration ǫ, those intervals can be omitted at error at most ǫ∥H∥.Piecewise continuity is enough to ensure the integrals exist.Source: Appendix B · p. 5
- Choosing the Gaussian width σ much smaller than T ∥H∥2 makes the difference between exact evolution and evolution with high-frequency components truncated negligible.The stated conclusion concerns the norm of X(∞).Source: Appendix B · p. 6
- For the two-level AC Stark example, the short-time approximation error scales as λ2T/ω, or ∥H∥2T σ, when the averaging window truncates the oscillating terms.The example and general bound reflect that high-frequency terms affect overall evolution only at second order in perturbation theory.Source: Appendix B · p. 6