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Simulation of Many-Body Hamiltonians using Perturbation Theory with Bounded-Strength Interactions
Sergey Bravyi, David P. DiVincenzo, Daniel Loss, Barbara M. Terhal
TL;DR
Perturbative simulation can appear to require a gap that grows with system size, limiting its use for physically scaled Hamiltonians. This paper uses Schrieffer-Wolff effective Hamiltonians and perturbation gadgets to map k-local targets to 2-local simulators. The simulator uses O(n) qubits and O(J) interaction strength while matching the target ground-state energy within εJn.
Problem
When ||V|| typically scales with n, requiring ||V|| ≪ Δ for perturbative convergence forces Δ to grow with system size.
Source: INTRODUCTION · p. 1Method
The paper derives low-energy effective Hamiltonians with the Schrieffer-Wolff transformation and combines perturbation gadgets to reduce interaction locality.
Source: INTRODUCTION · p. 2Results
For any fixed precision ε, a k-local target with interaction strength J has a 2-local simulator on O(n) qubits with interaction strength O(J) and ground-energy error at most εJn.
Source: INTRODUCTION · p. 2Takeaways & Limitations
The bounded simulator interaction strength removes the unphysical scaling of earlier constructions, at the price of an extensive absolute error that can be made arbitrarily small relatively for realistic Hamiltonians.
Source: INTRODUCTION · p. 2For generic perturbed Hamiltonians, cross-gadget terms may not be small, so their effect requires global analysis.
Source: INTRODUCTION · p. 3
Abstract
from arXiv · showhide
We show how to map a given n-qubit target Hamiltonian with bounded-strength k-body interactions onto a simulator Hamiltonian with two-body interactions, such that the ground-state energy of the target and the simulator Hamiltonians are the same up to an extensive error O(epsilon n) for arbitrary small epsilon. The strength of interactions in the simulator Hamiltonian depends on epsilon and k but does not depend on n. We accomplish this reduction using a new way of deriving an effective low-energy Hamiltonian which relies on the Schrieffer-Wolff transformation of many-body physics.
INTRODUCTION
The paper reduces bounded-strength k-local Hamiltonians to 2-local simulators while approximating the target ground-state energy with an extensive error. Its approach uses low-energy effective Hamiltonians derived through the Schrieffer-Wolff transformation and locally composed perturbation gadgets.
INTRODUCTION
- The approach is motivated by perturbation theory’s role in relating low-energy properties, while complexity theory uses a chosen target model to design a simpler simulator that preserves them.The paper expects similar reasoning for independent perturbation gadgets, but generic perturbed Hamiltonians may have cross-gadget terms that are not small and require global analysis.Source: INTRODUCTION · pp. 1, 3
- Requiring ||V|| ≪ Δ for perturbative convergence forces Δ to grow with n when ||V|| typically scales with n, making the simulator gap unphysical.The paper notes that generic perturbative expansions should not be expected to converge when ||V|| ≫ Δ.Source: INTRODUCTION · p. 1
- The paper develops rigorous low-order error bounds for effective-Hamiltonian expansions when local interaction strengths are small compared with Δ but ||V|| ≫ Δ.The formalism applies to k-local Hamiltonians in which each qubit participates in at most a constant number of interaction terms.Source: INTRODUCTION · p. 2
- For any fixed precision ε, a k-local target with interaction strength J has a 2-local simulator on O(n) qubits with interaction strength O(J) and ground-energy error at most εJn.The bounded interaction strength avoids the earlier constructions’ unphysical scaling; the extensive absolute error can be an arbitrarily small relative error for realistic Hamiltonians whose ground-state energy scales with nJ.Source: INTRODUCTION · p. 2
- The Schrieffer-Wolff transformation uses a block-off-diagonal anti-Hermitian generator to block-diagonalize the perturbed Hamiltonian and define its low-energy effective Hamiltonian.The effective Hamiltonian is the transformed Hamiltonian projected onto the low-energy subspace.Source: INTRODUCTION · p. 2
- Each perturbation gadget adds a mediator qubit whose low-energy effective Hamiltonian approximates a chosen k-body interaction.The mediator’s |1⟩ state is penalized, while the gadget perturbation is chosen to reproduce the target term in the |0⟩ subspace.Source: INTRODUCTION · pp. 2–3
- The construction combines local gadgets and uses a global unitary transformation to relate the simulator to the target with an O(εnJ) error.The mapping composes locality reduction by a factor of two with a separate 3-to-2-local gadget; the proof controls cross-gadget terms using gadget independence and block-off-diagonality.Source: INTRODUCTION · p. 3
- A lower-bound proof transforms each gadget Hamiltonian into an operator bounded below by the target interaction up to O(εJ), yielding a simulator ground-energy bound up to O(nεJ).The proof uses the large energy penalty in the high-energy block and small off-diagonal blocks after transformation.Source: INTRODUCTION · p. 3
I. THE GADGETS
The subdivision gadget and 3-to-2-local gadget use mediator qubits and Schrieffer-Wolff transformations to replace higher-locality interactions with lower-locality ones. Their perturbative transition paths generate the desired terms, while extra terms are canceled and truncation errors controlled.
Subdivision Gadget
- The subdivision gadget replaces JAB with a mediator-assisted interaction, using a second-order transition path to generate AB and Vextra to cancel unwanted A² and B² terms.A and B act on disjoint subsets of at most ⌈k/2⌉ qubits, and the simulator uses only (⌈k/2⌉+1)-body interactions.Source: §I · p. 4, Figure 1 · p. 5
- The subdivision gadget’s projected Schrieffer-Wolff Hamiltonian approximates JAB with error O(εJ) when Δ = Jε^-2.The transformation is truncated at first order; the projection removes the mediator’s high-energy block.Source: §I · p. 4
3-to-2-local Gadget
- The 3-to-2-local gadget encodes JABC using only 2-body interactions, with a third-order mediator-flip path generating the target term and Vextra canceling unwanted terms.The transition path is |0⟩u → |1⟩u → |1⟩u → |0⟩u.Source: §I · p. 4
- The 3-to-2-local gadget’s target-term error O(J^4/3Δ^-1/3) becomes O(εJ) by choosing Δ = Jε^-3.Its Schrieffer-Wolff transformation is truncated at third order.Source: §I · pp. 4–5
II. COMBINING THE GADGETS TOGETHER
The proof combines individually transformed gadget Hamiltonians while controlling interactions between gadgets. Local lower bounds and a projected-subspace upper bound together yield the claimed extensive ground-state-energy error.
Combining the gadgets together
- Summing the individual transformed-gadget inequalities yields H ≥ I ⊗ Htarget + O(εnJ), with I acting on the mediator qubits.The per-gadget inequality is Hu ≥ Iu ⊗ Htarget,u + O(εJ).Source: §II · p. 5
- The proof uses S = Σu Su to transform the full simulator and bounds cross-gadget terms using the fact that each Su flips its own mediator and local gadgets act independently.For both gadget types, the subdivision and 3-to-2-local analyses bound the remaining cross-talk contributions before establishing the projected-block error.Source: §II · p. 5
- Combining the lower bound from summed gadget Hamiltonians with the projected-subspace upper bound gives |λ(H) − λ(Htarget)| = O(εnJ).Each gadget contributes a local error O(εJ), while projection onto all mediators in |0⟩ gives the upper bound.Source: §II · p. 5
III. SIMULATION OVERHEAD
Recursive subdivision bounds the simulator interaction growth as a function of precision and locality. The stated overhead grows exponentially in k, with the authors noting that a direct gadget may improve that scaling.
Simulation overhead
- A direct k-to-2-local gadget may improve the exponential scaling in k.This is stated as a possibility, not a demonstrated improvement.Source: §III · p. 6
APPENDIX
The appendix bounds the transformed Hamiltonian using local commutators, then integrates incremental bounds and applies them recursively.
APPENDIX
- For H(t) = e^St H e−St, the increment in r_k is bounded by δt r_{k−1}(L(H), t) plus O((δt)^2).When k = 1, there are no O(δt) terms on the right-hand side of the transformation increment.Source: APPENDIX · pp. 6–7
- Continuity permits integrating the incremental inequality, and recursively applying the resulting bound proves the lemma.The recursion uses r_0(H, s) = ∥H∥.Source: APPENDIX · p. 7
- Because commutators of O(1)-local operators remain O(1)-local, only O(n) nonzero order-k elementary commutators contribute to L_k(H).Each qubit is acted on by O(1) operators S_i and H_j.Source: APPENDIX · p. 7
SYSTEMATIC SOLUTION FOR THE SCHRIEFFER-WOLFF TRANSFORMATION
The Schrieffer-Wolff equation has a unique formal Taylor-series solution, and the gadgets use truncated versions of that series.
SYSTEMATIC SOLUTION FOR THE SCHRIEFFER-WOLFF TRANSFORMATION
- The equation P e^S(H0 + βV)e^−S Q = 0 has a unique solution as a formal Taylor series in β.Its coefficients S_p can be computed straightforwardly.Source: SYSTEMATIC SOLUTION FOR THE… · p. 7
- The subdivision gadget truncates S at first order, while the 3-to-2-local gadget truncates it at third order.Source: SYSTEMATIC SOLUTION FOR THE… · p. 7