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Shortcuts to Adiabaticity in Digitized Adiabatic Quantum Computing
Narendra N. Hegade, Koushik Paul, Yongcheng Ding, Mikel Sanz, F. Albarrán-Arriagada, Enrique Solano, Xi Chen
TL;DR
Adiabatic evolution can require times that exceed current quantum computers’ coherence times, limiting fidelity. This paper applies digitized CD driving, including approximate local terms, to Ising spin systems. It reports improved final-state probabilities and fewer Trotter steps than digitized adiabatic evolution, alongside experimental two-spin fidelity around 0.93.
Problem
Adiabatic evolution can take longer than current quantum computers’ coherence times, while exact CD terms for complex many-body systems are difficult to calculate and implement.
Source: §I · p. 1, §IV · p. 5Method
Results
Around 0.93 fidelity is achieved experimentally for the two-spin target with the CD term, and the method reaches the final ground state with very few Trotter steps compared with digitized adiabatic evolution.
Source: §1 · p. 4, §IV · p. 6Takeaways & Limitations
The authors report that STA can reduce computational cost and achieve desired results within the device coherence time in their superconducting-processor study.
Source: §V · p. 7For complex many-body systems, exact CD terms can be difficult to calculate and implement, and the local variational terms are not useful for preparing entangled states in the described setting.
Source: §IV · p. 5, §2 · p. 5
Abstract
from arXiv · showhide
Shortcuts to adiabaticity are well-known methods for controlling the quantum dynamics beyond the adiabatic criteria, where counter-diabatic (CD) driving provides a promising means to speed up quantum many-body systems. In this work, we show the applicability of CD driving to enhance the digitized adiabatic quantum computing paradigm in terms of fidelity and total simulation time. We study the state evolution of an Ising spin chain using the digitized version of the standard CD driving and its variants derived from the variational approach. We apply this technique in the preparation of Bell and Greenberger-Horne-Zeilinger states with high fidelity using a very shallow quantum circuit. We implement this proposal in the IBM quantum computer, proving its usefulness for the speed up of adiabatic quantum computing in noisy intermediate-scale quantum devices.
I. INTRODUCTION
Digitized adiabatic quantum computation offers circuit-model flexibility, but adiabatic evolution can take longer than current devices’ coherence times. This work applies shortcut-to-adiabaticity driving to Ising systems, including approximate local terms and entangled-state preparation.
I. INTRODUCTION
- DAdQC combines adiabatic computation with the circuit model, whose flexibility supports arbitrary interactions and is consistent with error correction.Adiabatic computation evolves from an easily prepared initial ground state to a final ground state encoding the optimization solution.Source: §I · p. 1
- Adiabatic evolution time grows as the minimum energy gap shrinks with system size, potentially exceeding current devices’ coherence times and reducing fidelity.The adiabatic theorem requires evolution from the initial ground state to the final ground state.Source: §I · p. 1
- The paper applies STA to Ising spin systems, progressing from a single spin to interacting spins and using local CD terms to target smaller systems in few time steps.Exact CD terms are difficult to implement for many-body systems.Source: §I · p. 1
- Nested-commutator approximate CD terms are used to study a non-integrable Ising model and prepare Bell and GHZ states in larger systems.The paper also describes variational approaches to approximate CD driving and entangled-state preparation.Source: §I · p. 2
II. SINGLE SPIN SYSTEM
For a single spin, the paper digitizes adiabatic evolution with a CD term that vanishes at the endpoints. In the reported experiment, CD driving raises the final-state probability at T = 1, while longer circuits incur greater gate-error effects.
II. SINGLE SPIN SYSTEM
- The single-spin Hamiltonian interpolates between hxσx and hzσz through λ(t) = sin^2(ωt), with ω = π/2T.The initial and target states are |+⟩ and |1⟩; the CD term is constrained to vanish at both endpoints.Source: §II · p. 2
- First-order Trotter-Suzuki digitization uses n = T/∆t steps and has error of order O(∆t^2).Each Trotter step is implemented with three rotation gates.Source: §II · p. 2, Figure 1 · p. 2
- 0.997 versus 0.561: at T = 1, the experimental probability of obtaining |1⟩ is around 0.997 with CD driving and around 0.561 without it.The comparison uses ∆t = 0.2; the paper also reports nearly unit fidelity for STA at T = 1 with ∆t = 0.02.Source: §II · p. 3, Figure 2 · p. 3
- At T = 5, the adiabatic case averages around 0.927 fidelity, while increasing circuit depth raises gate errors and affects STA more than the adiabatic case.The experimental values differ slightly from exact simulation values, with a slightly larger difference for STA.Source: §II · p. 3
III. LOCAL COUNTER-DIABATIC DRIVING
The paper extends digitized adiabatic evolution to interacting Ising spins and considers local CD driving as an approximation for many-spin systems. This approach replaces whole-system control with interactions designed to control spins individually.
III. LOCAL COUNTER-DIABATIC DRIVING
- The many-spin setup is a one-dimensional lattice with time-dependent Ising exchange interaction J(t) and a rotating magnetic field.The initial spins align along the transverse field, and the final state is represented by an Ising Hamiltonian.Source: §III · p. 3
- Exact CD driving becomes more complex beyond two- and three-level systems, motivating local approximate interactions for interacting spins.The paper presents local driving as more general and extendable to larger spin counts.Source: §III · p. 3
- Local CD driving controls spins individually rather than acting on the whole system.The evolution uses the same scheduling choice as the preceding section.Source: §III · p. 3
1. Local CD from Berry’s algorithm
The local CD construction approximates each spin’s effective field and sums the resulting local driving terms. In the two-spin experiment, CD driving reaches high fidelity, but longer runs lose fidelity as gate errors and coherence limits accumulate.
1. Local CD from Berry’s algorithm
- The local field approximation combines hx(1 − λ(t)) with an effective longitudinal field containing the local z field and J0.The local CD driving is calculated for each spin and summed to obtain the modified evolution.Source: §1 · pp. 3–4
- The two-spin implementation uses |++⟩ as the initial state and |11⟩ as the target, with entangling and single-qubit rotation gates.The stated parameters are h0 = −1, h1 = h2 = 1, and J0 = −0.1.Source: §1 · p. 4
- Around 0.93 fidelity is achieved experimentally with the CD term for the two-spin target state, and the term is most suitable when evolution time T is small.In the ideal digital simulator, the target state can be achieved with almost unit fidelity; CD also reaches it faster than adiabatic evolution in a single evolution.Source: §1 · p. 4, Figure 3 · p. 4
- Increasing T raises circuit depth, and the resulting gate errors and limited coherence time gradually reduce fidelity.The paper attributes the decline to the increasing number of gates required to implement the CD term.Source: §1 · p. 4
2. Local CD from variational approach
The variational approach constructs local counter-diabatic driving with a y-direction field on each spin and analytically determines its coefficient by minimizing an action. It achieves high final-state probabilities in short digitized evolutions, while the single-spin approximation is limited when interactions dominate or the target is highly entangled.
2. Local CD from variational approach
- The variational method chooses a local y-direction adiabatic gauge potential and analytically calculates each coefficient α_j(t) by minimizing the action.Its local CD Hamiltonian is built from this gauge-potential ansatz.Source: §2 · p. 4
- For T = 1, the ideal simulator reaches almost unit final-state probability in five Trotter steps, especially when |J0| is small.The ibmqx2 implementation differs from the simulator because of device errors.Source: §2 · p. 4, Figure 4 · p. 5
- For J0 ≪ h_j^z, both local CD methods yield around 98% ground-state probability, while the Berry-formula method falls to 35% even for two qubits as coupling grows.The variational approach has significantly higher probability at large |J0|; for a degenerate ground state, CD drives the system to one of its eigenstates.Source: §2 · p. 5, Figure 4 · p. 5
- The single-spin approximation undermines interaction effects and does not reach the exact ground state when J0 is comparable to or stronger than the local field.For highly entangled targets, the authors say the CD term must incorporate spin-spin couplings rather than only single-spin σy driving.Source: §2 · p. 5
IV. APPROXIMATE COUNTER-DIABATIC DRIVING
The section develops variational approximate counter-diabatic driving for Ising systems, then evaluates it in degenerate-state and Bell/GHZ preparation. The reported results include high experimental fidelities for small states and improved many-spin fidelity from higher-order commutators.
Variational approximate CD terms
- Because exact many-body CD terms are difficult and can be nonlocal, a variational nested-commutator expansion provides approximate gauge potentials with order l controlling the expansion.The approach minimizes an action to obtain the approximate gauge potential.Source: §IV · pp. 5–6
Nested-commutator driving
- For a degenerate final state, first-order NC driving reaches the target in very few Trotter steps and has much better fidelity than local CD.The figure plots fidelity against evolution time for local CD and first-order NC ansatz on ibmq vigo.Source: §IV · p. 6, Figure 5 · p. 6
Many-spin GHZ preparation
- The many-spin GHZ study applies the variational procedure to an Ising spin chain with periodic boundary conditions.Source: §IV · p. 6
Bell and GHZ state preparation
- For two spins, the first-order variational CD term is exact in ideal conditions; the section also specifies coefficients for Bell and three-qubit GHZ preparation.The experimental demonstration uses two- and three-qubit systems.Source: §IV · p. 6
- Experimental fidelity was 0.984 for the Bell state and 0.819 for the GHZ state, versus ideal digital fidelities of 0.999 and 0.935, respectively.The final-state density matrices were obtained by tomography.Source: §IV · p. 6, Figure 6 · p. 6
Many-spin GHZ preparation
- With six Trotter steps, first-order CD fidelity declines as GHZ system size grows, while higher-order commutators can overcome that decline.The comparison uses an ideal digital simulator and includes the naive approach without CD.Source: §IV · p. 7, Figure 7 · p. 7
V. CONCLUSION
The authors demonstrate digitalized shortcuts to adiabaticity on a superconducting quantum processor using local and approximate counter-diabatic driving for a one-dimensional Ising spin chain. They report fewer time steps than digitized adiabatic quantum computing, high-fidelity Bell and GHZ preparation, and a gate-count trade-off for higher-order many-qubit terms.
Conclusion
- Digitalized STA was demonstrated on a superconducting quantum processor for a one-dimensional Ising-chain problem using local and approximate CD driving.The approximate terms are calculated using Berry’s algorithm and a variational approach.Source: §V · p. 7
- Compared with DAdQC, STA-assisted evolution requires fewer time steps, which the authors report reduces losses from decoherence and accumulated gate errors.Source: §V · p. 7
- Local CD is effective for weakly interacting spin chains but not for strong interactions or degenerate targets; approximate NC driving addresses these cases.Source: §V · p. 7
- For many-qubit systems, higher-order terms can improve fidelity at the cost of more gates; the authors describe the work as a first successful STA realization on a contemporary superconducting circuit-based quantum computer.Source: §V · p. 7
Appendix A: Method of digitization
The appendix digitizes time-dependent Hamiltonian evolution by dividing the total time into equal steps and applying a first-order Trotter–Suzuki decomposition. It also describes how the resulting unitary operations are implemented in the circuit model and notes the approximation’s error sources.
Appendix A: Method of digitization
- The circuit model simulates a time-dependent Hamiltonian represented as a sum of M k-local terms, each acting on at most k qubits.Source: Appendix A · p. 7
- The first-order Trotter–Suzuki decomposition divides total evolution time T into n equal steps of width Δt, with error of order O(Δt^2).Higher-order decompositions can further reduce the approximation error.Source: Appendix A · p. 7
- For CD-driven digital evolution, the simulation can depend on the number of Trotter steps rather than total time T, while ignoring within-step Hamiltonian variation adds an error of about ∥∂H/∂t∥Δt per step.Source: Appendix A · p. 8
- The resulting unitary operators are implemented as circuit gates, with k-body unitaries decomposable into single- and two-qubit operations.The appendix illustrates a one-step decomposition and the variables representing changes in the Hamiltonian.Source: Appendix A · p. 8
Appendix B: Approximate CD term using variational method
Counter-diabatic driving adds an auxiliary term to the original Hamiltonian, but the exact gauge potential is impractical for many-body spin systems because it entails rapidly growing gate counts. The variational approach uses approximate gauge potentials, with nested commutators providing a more general ansatz when a simple single-qubit approximation fails.
Appendix B: Approximate CD term using variational method
- Counter-diabatic driving adds an auxiliary term to the original Hamiltonian and evolves the system with an effective Hamiltonian.Source: Appendix B · p. 8
- The exact gauge potential contains non-local many-body interactions whose gate requirements grow rapidly with system size.The paper therefore uses an approximate gauge potential for practical implementation.Source: Appendix B · p. 8
- The variational method chooses the approximate gauge potential by minimizing its operator distance from the exact gauge potential, equivalently minimizing the action.Source: Appendix B · p. 8
- The single-qubit approximation works well for many-body systems but fails when spin interactions lead the adiabatic gauge potential.Source: Appendix B · p. 8
- A nested-commutator ansatz generalizes the approximation and approaches the exact gauge potential as its order tends to infinity.The nested-commutator sequence is proposed in.Source: Appendix B · p. 8
Appendix C: Error Analysis
The error analysis identifies discretization, cumulative gate, and measurement errors, and notes that long evolutions also introduce relaxation and dephasing concerns. Smaller Trotter steps reduce discretization error but require more steps and gates; cross-talk and other environmental effects are not included in the simulations.
Appendix C: Error Analysis
- Table II compares the quantum-gate counts for digital adiabatic evolution with and without the CD term.Source: Table II · p. 9
- Cumulative gate error combines single-qubit and CNOT errors and increases linearly with circuit depth; measurement error arises from end-of-evolution measurements.Source: Appendix C · p. 9
- Cross-talk and other environmental effects can disturb the simulation but were not considered.Long evolution also requires considering energy relaxation and dephasing.Source: Appendix C · p. 9
- First-order Trotter-Suzuki simulation has error O(∆t^2), while reducing ∆t requires more steps and increases total gate count.For a fixed total time, smaller ∆t can reduce Trotter error.Source: Appendix C · p. 9