Source-linked AI summary
Programmable Superpositions of Ising Configurations
Lukas M. Sieberer, Wolfgang Lechner
TL;DR
The paper addresses how to prepare superpositions of many-body bit-string configurations with deterministically programmable probabilities. It encodes the configurations in a lattice-gauge Ising ground-state manifold, tunes four-body constraint strengths during adiabatic-diabatic evolution, and derives an effective model for efficient control. The framework produces programmable amplitudes, with iterative optimization achieving nearly exact target probabilities in the reported example.
Problem
Degenerate-ground-state adiabatic preparation can produce exponentially biased populations, leaving deterministic programming of individual configuration probabilities unresolved.
Source: §I · p. 1Method
The framework encodes polynomially many bit strings in a lattice-gauge Ising ground-state manifold and tunes gauge constraint strengths using controlled adiabatic-diabatic dynamics and an effective model.
Source: §I · p. 1, §II · p. 2, C. Calculation of the effective… · p. 4Results
The reported three-state example reaches final probabilities almost exactly 1/M after optimizing the constraint strengths, showing fidelity is limited by control-parameter accuracy rather than fundamentally.
Source: A. Preparation of superposition states · p. 7Takeaways & Limitations
Gauge degrees of freedom provide programmable control over superposition weights while preserving the encoded low-energy subspace.
Source: §II · p. 2, §V · p. 9
Abstract
from arXiv · showhide
We present a framework to prepare superpositions of bit strings, i.e., many-body spin configurations, with deterministic programmable probabilities. The spin configurations are encoded in the degenerate ground states of the lattice-gauge representation of an all-to-all connected Ising spin glass. The ground state manifold is invariant under variations of the gauge degrees of freedom, which take the form of four-body parity constraints. Our framework makes use of these degrees of freedom by individually tuning them to dynamically prepare programmable superpositions. The dynamics combines an adiabatic protocol with controlled diabatic transitions. We derive an effective model that allows one to determine the control parameters efficiently even for large system sizes.
I. INTRODUCTION
The paper proposes programmable superpositions of a polynomial number of bit strings by encoding them in degenerate ground states and tuning lattice-gauge constraints during adiabatic-diabatic evolution.
I. INTRODUCTION
- The framework targets deterministic programming of probabilities for a polynomial number M of bit strings within the 2^N-dimensional Hilbert space.Degenerate-ground-state adiabatic preparation alone can produce exponentially biased populations, motivating explicit probability control.Source: §I · p. 1
II. PROTOCOL
- Bit strings are encoded as degenerate ground states of an all-to-all Ising model, then mapped to a lattice-gauge model with four-body parity constraints.The mapping uses K = N(N −1)/2 physical spins, while the constraints remove additional gauge degrees of freedom.Source: §I · p. 1, §II · p. 2
- Individually tuning the constraint strengths Cp leaves the final low-energy subspace unchanged while shaping the quantum paths and programmable squared amplitudes.The protocol initializes a transverse-field ground state and evolves toward a superposition of the encoded configurations.Source: §II · p. 2
- The protocol stores the selected classical configurations and their probabilities in a quantum state |ψ⟩ = Σ_n a_n|x_n⟩ with p_n = |a_n|^2.The encoded bit strings are product states in the σz basis, and only a polynomial number of configurations is selected from the full space.Source: §II · p. 2
III. EFFECTIVE MODEL
The effective-model section reduces the preparation dynamics to the lowest-energy instantaneous manifold, where controlled diabatic evolution determines the final superposition.
A. Adiabatic-diabatic dynamics
- Up to the freeze time td evolution is adiabatic, while afterward the state is approximately frozen, reducing probability control to choosing Cp so the instantaneous ground state has the desired composition.This approximation gives |ψ(T)⟩ ≈ |ϕ1(td)⟩ up to phases irrelevant to probabilities.Source: A. Adiabatic-diabatic dynamics · p. 3
- The M lowest instantaneous eigenstates form an adiabatic manifold whose separation from higher states suppresses leakage when the protocol run time T is sufficiently large.The manifold becomes degenerate near the end, so its internal dynamics is intentionally diabatic rather than fully adiabatic.Source: B. Derivation of the effective model · p. 3, A. Adiabatic-diabatic dynamics · p. 3
B. Derivation of the effective model
- A Schrieffer-Wolff transformation decouples the adiabatic manifold from excited states, after which the reduced effective Hamiltonian is diagonalized within the M-dimensional space.This construction avoids directly diagonalizing the full Hamiltonian and supplies the instantaneous states governing the reduced dynamics.Source: B. Derivation of the effective model · p. 3
- The simplest effective model drops a time-derivative contribution because it is suppressed by 1/T, while retaining that term can improve accuracy straightforwardly.The approximation is therefore controlled by the protocol run time and can be systematically refined.Source: B. Derivation of the effective model · pp. 3–4
C. Calculation of the effective Hamiltonian
The effective Hamiltonian is constructed by decoupling the adiabatic manifold from excited states with a Schrieffer-Wolff transformation. Its perturbative matrix elements show how constraint strengths tune the composition of the instantaneous ground state and thereby the target superposition.
C. Calculation of the effective Hamiltonian
- The Schrieffer-Wolff transformation decouples the adiabatic manifold from excited states, incorporating virtual transitions into an effective Hamiltonian.The perturbative construction is used primarily near the diabatic time, where A(td)/Cp is small; the resulting dynamics remains non-perturbative.Source: C. Calculation of the effective… · p. 4
- The unperturbed Hamiltonian has the encoded states |zn⟩ as a degenerate ground-state manifold, while the transverse field supplies the perturbation.The ground-state projector and excited-state projector define the blocks that the transformation separates.Source: C. Calculation of the effective… · p. 4
- Single-spin flips generate diagonal energy corrections at second order and couple encoded states at an order set by their Hamming distance.Because a single flip leaves the ground-state manifold, each perturbative order also carries constraint-dependent energy denominators.Source: C. Calculation of the effective… · p. 4
- The fourth-order effective Hamiltonian is sufficient for examples with ground states separated by Hamming distances three and four, while larger distances require higher orders.The expansion separates diagonal and off-diagonal contributions, with a systematic iterative procedure available for higher orders.Source: C. Calculation of the effective… · p. 5
- The constraint strengths Cp enter the effective Hamiltonian and can be tuned so its instantaneous ground state at td equals the desired superposition.This tuning adjusts the composition of the effective eigenstates without changing the encoded ground-state manifold.Source: C. Calculation of the effective… · pp. 5–6
- The effective model also provides a computationally inexpensive framework for studying fair sampling of degenerate spin-glass ground states under quantum annealing.In the lattice-gauge formulation, the constraint strengths Cp act as additional tuning parameters for fair sampling.Source: C. Calculation of the effective… · p. 6
D. Estimation of td
The diabatic time td is estimated by treating each approaching pair of levels as an individual Landau-Zener problem. The earliest transition to become diabatic determines the time used for control-parameter selection.
D. Estimation of td
- Each pair of levels approaching degeneracy is characterized by a time-dependent velocity and gap, with the diabatic boundary set by vnn′/Δnn′^2 = π.The pairwise estimates are compared to select the earliest diabatic event.Source: D. Estimation of td · p. 6
- The smallest pairwise Landau-Zener time td identifies the transition that becomes diabatic first and is used to determine the constraint strengths.Transitions between states with larger Hamming distances generally have larger td.Source: D. Estimation of td · p. 6
E. Determination of control parameters
The control parameters are obtained by matching the effective-model ground state at td to the desired amplitudes and then refining them through time-evolution-based optimization. This procedure can achieve fidelity close to one while remaining efficient in the reduced M-dimensional space.
E. Determination of control parameters
- For a target superposition, the constraint strengths Cp are chosen so the effective-model ground state at td has the required amplitudes.The method minimizes a cost function comparing the target coefficients an with the effective-model coefficients bn.Source: E. Determination of control parameters · p. 6
- The approximate preparation can reach fidelity close to one, and iterative optimization further updates Cp using the full effective-model time evolution.The optimization starts at finite t0 and is largely insensitive to t0 for 0 < t0 ≲ td in the reported examples.Source: E. Determination of control parameters · p. 6
- Optimization remains efficient in the M-dimensional effective model even when the full physical Hilbert space is too large for classical simulation.The reported setting allows state preparation on a quantum device despite the intractability of simulating the total space classically.Source: E. Determination of control parameters · p. 6
IV. EXAMPLE
The example uses three four-bit configurations to demonstrate preparation of both uniform and nonuniform target probability distributions.
IV. EXAMPLE
- The example encodes three bit strings, x1 = 1011, x2 = 1100, and x3 = 1111, and targets both uniform and nonuniform probabilities.The nonuniform target is p1 = 0.2, p2 = 0.3, and p3 = 0.5, alongside the uniform choice pn = 1/M.Source: §IV · p. 6
A. Preparation of superposition states
The protocol encodes three bit strings as degenerate ground states and tunes their final probabilities through constraint strengths and controlled diabatic dynamics. The effective-model and exact results agree closely for both equal and unequal target probabilities.
A. Preparation of superposition states
- The lattice-gauge model uses constraint strengths C1, C2, and C3 to control a superposition of three degenerate ground states.The example has two three-body constraints and one four-body constraint, with the effective Hamiltonian determining how the parameters affect the dynamics.Source: A. Preparation of superposition states · pp. 6–7
- The optimized parameters produce nearly equal probabilities |a1|2 = 0.344, |a2|2 = 0.347, and |a3|2 = 0.309, while full-dynamics optimization yields probabilities almost exactly 1/M.For a second target, iterative optimization gives |a1|2 = 0.219, |a2|2 = 0.297, and |a3|2 = 0.484, again matching the approximate solution.Source: A. Preparation of superposition states · p. 7
- The time-dependent spectrum collapses onto the degenerate final states, with exact, effective-model, and improved iterative results in good agreement.The diabatic times are approximately td ≈0.5T for equal sampling and td ≈0.6T for the unequal target probabilities.Source: Figure 2 · p. 8, A. Preparation of superposition states · p. 8
- The adiabatic-diabatic protocol remains adiabatic within the instantaneous ground-state manifold until td, after which the state is effectively frozen while the Hamiltonian continues to evolve.The desired probabilities are therefore reached before the final time T.Source: A. Preparation of superposition states · p. 8
B. Robustness to constraint errors
The programmed probabilities are robust to errors in the constraint strengths. Around the optimized parameters, even substantial relative errors cause only modest probability deviations.
B. Robustness to constraint errors
- The solution is remarkably insensitive to control-parameter errors near the optimized values.This robustness is evaluated for equal target probabilities in the three-state example.Source: B. Robustness to constraint errors · p. 8
- A 20% error in the most sensitive constraint C1 changes the final probability by 10%, while errors in the other constraints cause smaller deviations.The figure varies the relative constraint error over e from 0.6 to 1.4 around the optimized point.Source: Figure 3 · p. 8, B. Robustness to constraint errors · p. 8
C. Ergodicity of the solution space
The protocol has more available constraint parameters than probability degrees of freedom for large systems, suggesting broad programmability. In the three-state example, systematic scanning finds an almost ergodic solution plane, while the conclusion identifies tunable weights as the central capability.
C. Ergodicity of the solution space
- For large systems, the available constraint parameters generally outnumber the M −1 independent probabilities, suggesting that arbitrary superpositions can be prepared.The number of constraints grows quadratically with the bit-string length N, whereas normalization leaves M −1 probability variables.Source: C. Ergodicity of the solution space · p. 8
- Scanning C1, C2, C3, and td for M = 3 produces an almost ergodically populated two-dimensional probability plane.The scan uses constraint values from 0.1 to 4.0 in steps of 0.1 and times spaced by T/30.Source: C. Ergodicity of the solution space · p. 8
- The authors expect full ergodicity for larger systems because they provide more degrees of freedom than the probability variables require.This is presented as an expectation based on the degree-of-freedom count.Source: C. Ergodicity of the solution space · pp. 8–9
V. CONCLUSIONS
- The framework controls dynamical preparation of superpositions with tunable weights and uses an effective multi-level Landau-Zener model to determine control parameters.The conclusion also identifies phase control as a future challenge beyond the demonstrated weight control.Source: §V · p. 9
Appendix A: Effective Hamiltonian for the example in Sec. IV
The appendix derives the example’s effective Hamiltonian by tracking how constraint strengths modify excitation energies and perturbative transitions between encoded ground states. Off-diagonal couplings are organized by spin-flip paths whose leading order is set by Hamming distance.
Appendix A: Effective Hamiltonian for the example in Sec. IV
- For the example, the effective Hamiltonian can be calculated numerically from Eq. (22), while explicit coefficients expose how the constraint strengths control the dynamics.The explicit coefficients are reported in Eqs. (27), (28), and (29).Source: Appendix A · pp. 9–10
- Diagonal effective-Hamiltonian elements are obtained at second order in perturbation theory from local fields and constraints involving each flipped spin.A flipped spin violates the constraints satisfied by the corresponding encoded ground state.Source: Appendix A · p. 9
- An off-diagonal coupling between encoded states is built from perturbative paths corresponding to different orders of the required single-spin flips.For states separated by Hamming distance hnn′, the leading contribution occurs at perturbative order hnn′.Source: Appendix A · pp. 9–10
- The constraint strengths Cp enter the effective Hamiltonian through energy denominators associated with violated constraints after spin flips.The example contains two three-body and one four-body constraint interactions.Source: Appendix A · pp. 9–10