Time-Optimal Two- and Three-Qubit Gates for Rydberg Atoms

Sven Jandura; Guido Pupillo

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Time-Optimal Two- and Three-Qubit Gates for Rydberg Atoms

Sven Jandura, Guido Pupillo

arXiv:2202.00903v2Feb 2, 2022quant-ph

TL;DR

The paper asks how to implement CZ and C2Z gates on Rydberg atoms with the shortest global laser pulses while addressing relevant fidelity limitations. It combines GRAPE with PMP to obtain smooth, few-parameter pulses, achieving short gate durations and theoretical fidelities compatible with error correction under considered conditions.

Problem
Rydberg-atom CZ and C2Z gates require fast, high-fidelity pulses despite blockade, decay, and other experimental error sources.
Method
The authors combine GRAPE's numerical pulse optimization with PMP to derive smooth pulse Ansätze using only a few variational parameters.
Results
TΩmax = 7.612 for CZ and TΩmax = 16.43 for C2Z, with global pulses improving upon the cited traditional and prior global implementations.
Takeaways & Limitations
Global CZ operation requires no single-site addressability, while short pulses mitigate duration-dependent errors and the reported fidelities are compatible with error correction.
Takeaways & Limitations
The study considers gates using one Rydberg state per atom in the blockade regime and does not explicitly treat atom-motion coupling.

Abstract

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We identify time-optimal laser pulses to implement the controlled-Z gate and its three qubit generalization, the C$_2$Z gate, for Rydberg atoms in the blockade regime. Pulses are optimized using a combination of numerical and semi-analytical quantum optimal control techniques that result in smooth Ansätze with just a few variational parameters. For the CZ gate, the time-optimal implementation corresponds to a global laser pulse that does not require single site addressability of the atoms, simplifying experimental implementation of the gate. We employ quantum optimal control techniques to mitigate errors arising due to the finite lifetime of Rydberg states and finite blockade strengths, while several other types of errors affecting the gates are directly mitigated by the short gate duration. For the considered error sources, we achieve theoretical gate fidelities compatible with error correction using reasonable experimental parameters for CZ and C$_2$Z gates.

1 Introduction

This section motivates faster, higher-fidelity Rydberg-atom gates and identifies global, time-optimal pulses for CZ and C2Z operations. The approach combines numerical and semi-analytical optimal-control methods to simplify pulse descriptions and address experimental error sources.

High-fidelity two-qubit gates support deep noisy circuits and fault-tolerant quantum computing, while k-qubit gates can reduce algorithmic gate counts.
Finite blockade strength, Rydberg decay, intermediate-state scattering, phase noise, intensity variation, and Doppler shifts limit experimental gate fidelities.The paper notes that optimal-control methods can improve speed and fidelity despite these error sources.
Time-optimal pulses are desirable because shorter durations mitigate several errors, while global control avoids single-site addressing and simplifies implementation.The requirements emphasized for neutral-atom gates include speed, global control, and robustness.
The time-optimal global CZ pulse keeps laser amplitude constant while varying phase smoothly, is about 10% faster than Ref., and remains optimal with single-site addressability.This shows that individual addressing is unnecessary for the CZ gate's speed optimum.
For C2Z, two qualitatively different global pulses differ by less than 1% in speed and are faster than the single-site-addressable pulse in Ref. [20].The paper identifies these as the first time-optimal C2Z pulses to its knowledge.
GRAPE locates optimal pulses using many parameters, after which PMP fits them with smooth pulse descriptions requiring only 4 parameters for CZ and 6 for C2Z.The combined procedure reduces the parameterization while retaining the optimized pulse structure.

2 Theoretical Tools

The paper models Rydberg-mediated phase gates with block-diagonal Hamiltonians and evaluates them using averaged fidelity. These tools characterize CZ and C2Z implementation and expose pulse symmetries that generate equivalent gate constructions.

Each atom is modeled as a three-level system with qubit states |0⟩ and |1⟩ plus a Rydberg state |r⟩, with lasers coupling |1⟩ to |r⟩.The cases n = 2 and n = 3 describe the two- and three-qubit gates studied.
The blockade-strength interaction shifts the energy of states in which two atoms occupy the Rydberg state.This interaction is represented by the term B_jk |rr⟩_jk⟨rr| in the Hamiltonian.
Because the Hamiltonian is block-diagonal in subspaces indexed by computational bit strings, each basis-state evolution can be solved in a reduced Hilbert space.This reduction simplifies the GRAPE and PMP calculations.
The model implements phase gates, with CZ and C2Z identified by specific combinations of conditional phases equal to π.For CZ, the condition is ξ11 − ξ01 − ξ10 = π; C2Z additionally requires the stated three-qubit phase relations.
The averaged fidelity measures agreement with the desired phase gate over normalized computational-subspace states, and gate error is defined as 1 − F.A single unnormalized propagated state can be sufficient because of Hamiltonian block-diagonality.
Phase shifts, time reversal, and complex conjugation generate alternative pulses with related phases, durations, or blockade strengths.These operations show that a time-optimal implementation need not be unique.

2.2 Quantum Optimal Control Methods

The control strategy combines GRAPE's numerical search with PMP's low-dimensional optimality conditions. GRAPE finds candidate time-optimal pulses, while PMP reconstructs them from a small set of initial costates.

GRAPE Algorithm: GRAPE optimizes piecewise-constant controls by minimizing a terminal objective through gradient-based numerical optimization.The implementation uses several hundred pieces and the BFGS gradient-descent algorithm.
Pontryagin’s Maximum Principle: PMP supplies necessary conditions for time-optimal trajectories and can reduce an infinite-dimensional control search to initial costates.It does not itself provide a universal algorithm for finding the optimal parameters.
Pontryagin’s Maximum Principle: For the Schrödinger equation, PMP represents costates that evolve under the same Hamiltonian structure as the quantum states.The complex costate combines real and imaginary costates into |χ⟩.
Pontryagin’s Maximum Principle: When the control supremum is unique, initial states and costates determine the full optimal trajectory.This is the condition enabling the low-dimensional representation.
Combined GRAPE–PMP approach: The combined procedure reconstructs GRAPE pulses from initial costates using 4 parameters for CZ and 6 for C2Z.The resulting control landscapes are four- and six-dimensional, respectively.

3 Time-Optimal Gates at Infinite Blockade Strength (B = ∞)

At infinite blockade strength, GRAPE identifies smooth global pulses implementing CZ and C2Z gates at minimum duration. The CZ pulse is about 10% faster than a prior pulse, while single-site addressability provides no further speedup.

3.1 Global CZ Gate: The CZ pulse uses maximal amplitude throughout, with its phase optimized as a smooth time-dependent function.GRAPE optimizes the laser phase and single-qubit phase after setting |Ω(t)| = Ωmax.
3.2 CZ Gate with single-site addressability: Single-site addressability does not reduce the minimum CZ gate error, so the time-optimal individually addressed pulse is identical to the global pulse.The non-global optimization uses independent complex controls Ω1(t) and Ω2(t).
3.3 Global C2Z Gate: The global C2Z pulses implement the three-qubit phase gate using the |001⟩, |011⟩, and |111⟩ sectors and smooth laser controls.Up to single-qubit z rotations, the target action is |xyz⟩↦(−1)^xyz |xyz⟩.
3.3 Global C2Z Gate: For C2Z, GRAPE reaches gate errors 1−F ≲10−12 near TΩmax ≃16.5, with two distinct near-optimal pulse families.The two pulses have similar durations but qualitatively different phase trajectories.

4 Semi-Analytical Description of the Pulses using the PMP

The PMP converts GRAPE-discovered piecewise-constant controls into smooth time-optimal pulses governed by ordinary differential equations. This reduces the pulse descriptions from hundreds of parameters to 4 for CZ and 6 for C2Z.

GRAPE-to-PMP reduction: GRAPE first searches over hundreds of pulse parameters, while PMP fits the resulting controls using a time-optimal differential equation.The reconstructed PMP pulses agree closely with the GRAPE pulses.
Reduced parametrization: The PMP reduces the CZ and C2Z control landscapes to 4 and 6 variational parameters, respectively.The reduced parameters are initial costates together with pulse duration.
Continuous optimal control: Because the Hamiltonian depends nonlinearly on the laser phase, the optimal control is continuous rather than bang-bang.The phase is obtained from the PMP maximization condition.
Pulse reconstruction: Given the initial costates, the Schrödinger and costate equations reconstruct the entire pulse, provided A2 + B2 does not vanish during evolution.The final duration and costates are optimized against the gate error.
Validation: The PMP reconstruction agrees excellently with GRAPE, showing that the discretized controls approximate simple smooth pulses.The agreement is demonstrated for the CZ and C2Z pulses.

5 Minimizing the Decay of the Rydberg State

Rydberg-state decay is modeled by a non-Hermitian loss term, making gate error proportional to the average Rydberg-state dwell time. GRAPE therefore optimizes this dwell time while preserving unit fidelity in the decay-free limit.

Decay model: Adding the loss term −iΓ|r⟩⟨r|/2 gives 1−F = ΓTR for a decay-free exact gate.TR is the average time atoms spend in the Rydberg state, and the model slightly overestimates error by neglecting repopulation after decay.
Optimization target: GRAPE minimizes TR over pulses that implement CZ or C2Z with fidelity 1 when Γ = 0.The optimization uses a small positive decay rate so the resulting pulses remain nearly perfect without decay.
CZ decay mitigation: For CZ, the time-optimal pulse is essentially also the minimum-TR pulse, with TRΩmax = 2.957 versus 2.947 for the optimized low-TR pulse.The comparison low-TR pulse has duration TΩmax = 30.
C2Z decay mitigation: For C2Z, minimizing TR improves on both time-optimal pulses, whose dwell times are TRΩmax = 6.90 and 7.52.Thus duration optimality and Rydberg-exposure optimality coincide for CZ but differ for C2Z.

6 Gates at Finite Blockade Strength

At finite blockade strength, CZ pulses remain close to the infinite-blockade solutions, while C2Z admits both a much faster but blockade-sensitive pulse and slower robust alternatives. Longer optimized pulses reduce blockade-induced errors by modulating amplitude according to populations of ac-Stark-shifted states.

6.1.2 Speeding up the C2Z gate: At B = 10Ωmax, a C2Z pulse with TΩmax = 10 achieves gate error 1−F = 3·10^-10 by significantly populating doubly excited Rydberg states.Rapid phase variation produces detuning of order B, indicating operation in the Rydberg antiblockade regime.
6.1.2 Speeding up the C2Z gate: Reducing B by 10% raises the fast C2Z pulse’s gate error to 1−F = 0.59, demonstrating strong sensitivity to blockade-strength variations.The speedup relies on accessing states with two atoms in the Rydberg state, which is unavailable at infinite blockade.
6.1.3 Pulses for the C2Z Gate resembling the B = ∞ pulses: C2Z pulses close to the infinite-blockade solutions achieve 1−F < 10^-9 at B = 10Ωmax, with durations TΩmax = 16.4, 16.6, 17.1, and 16.2.When B is reduced by 10%, their gate error never exceeds 4·10^-4, an improvement of three orders of magnitude over the faster pulse.
6.1.3 Pulses for the C2Z Gate resembling the B = ∞ pulses: For linear atom arrangements, no pulses close to the infinite-blockade solutions attain F = 1 under the stated approximation and B ≫Ωmax limit.Different atom pairs experience different finite-blockade effects under the same global laser.
6.2 Variable Blockade Strength: For both gates, longer pulses reduce finite-blockade effects beyond simple stretching by modulating laser amplitude to limit ac-Stark shifts during high population.For CZ, the rescaled error approaches α ≈28; for C2Z, it approaches α ≈1300.

7 Gate Errors for a Specific Setup

The paper evaluates time-optimal CZ and C2Z pulses under a specific cesium experimental setup, combining Rydberg decay and finite-blockade errors. Optimizing the peak Rabi frequency balances these two error sources and yields low theoretical gate errors.

A 540µs Rydberg-state lifetime is assumed for cesium in the |107P3/2, mj = 3/2⟩ state at 300K.
Optimizing Ωmax while fixing dimensionless pulse durations balances Rydberg decay against finite-blockade errors.
At B/2π = 3GHz, minimum errors are 7.0 · 10^-5 for CZ and 2.8 · 10^-4 for C2Z.
At B/2π = 180MHz, errors remain 4.6 · 10^-4 for CZ and 1.8 · 10^-3 for C2Z.
Moderate blockade and Rabi frequencies suffice for errors of order 1 − F ≲ 10^-3 for both gates.

8 Conclusion

The work identifies time-optimal global pulses for CZ and C2Z gates in the Rydberg blockade regime. The pulses are short, smoothly parameterized, experimentally simpler for global control, and achieve low modeled errors under the considered conditions.

The time-optimal global pulse durations are TΩmax = 7.612 for CZ and TΩmax = 16.43 for C2Z.These improve on the traditional non-global pulses and the previously reported global CZ pulse with TΩmax = 8.585.
Global pulses address all gate atoms with one laser, eliminating the need for single-site addressability.For CZ, single-site addressability provides no speedup over the time-optimal global pulse.
GRAPE combined with PMP reduces the pulse descriptions to 4 parameters for CZ and 6 for C2Z.The PMP description enables reproducing the pulses from the parameters in Table 1 without rerunning GRAPE.
Robustness was optimized against Rydberg decay and finite blockade using fixed-blockade compensation and large-blockade robustness strategies.The fixed-blockade approach works best when B is known exactly, whereas the second works well across large blockade strengths.
Modeled errors reach 7 · 10^-5 for CZ and 3 · 10^-4 for C2Z at B/2π = 3GHz and Rabi frequencies near 10MHz.At B/2π = 180MHz, the reported errors are 4.6 · 10^-4 and 1.8 · 10^-3, respectively.
The study considers gates using one Rydberg state per atom and operating in the blockade regime.The paper notes that more robust gates may use several Rydberg states, while ultrafast gates may operate outside the blockade regime.

A Estimation of the Effective Blockade Strength

The appendix estimates the effective blockade strength for cesium by diagonalizing a multistate Rydberg Hamiltonian. Resonances produce a sharp, nonmonotonic reduction at short distances, limiting the usable maximum to 180MHz near 7µm.

The calculation uses cesium atoms in |107p3/2, m = 3/2⟩ and |6S1/2, F = 4, mF = 4⟩, aligned perpendicular to the quantization axis.
The two-atom Hamiltonian includes Rydberg-state binding energies and dipole-dipole couplings between pair states.The single-atom states are labeled a, b, c, and d, with Ea and Eb denoting their binding energies.
Diagonalization includes Rydberg states with 104 ≤ n ≤ 110 and l ≤ 3, using eigenstate overlap with |rr⟩ to determine effective blockade behavior.
At R ≈ 6.5µm, resonant eigenstates cause the effective blockade strength to drop sharply.The resonant states are primarily composed of 106S1/2 + 109S1/2 configurations and each has squared overlap 0.02 with |rr⟩.
The maximal usable effective blockade is Beff/2π = 180MHz at R ≈ 7µm, restricting analysis to R > 6.5µm.

B Gate Error due to Decay

The appendix derives the gate error caused by Rydberg decay for an ideal phase gate. In the small-decay limit, the error is proportional to the decay rate and the average time spent in the Rydberg state.

The derivation starts from an ideal phase-gate pulse with fidelity F = 1 in the absence of decay.
For ΓTR ≪ 1, the resulting gate error is 1 − F = ΓTR.TR is the average time spent in the Rydberg state.
Rydberg decay is represented by adding the non-Hermitian term −iΓΠr/2 to the Hamiltonian.The projector Πr selects states containing Rydberg excitations; for B = ∞ it projects onto states with exactly one Rydberg atom.
Expanding the time-evolution operator to first order in Γ yields the decay correction to the gate dynamics.
Using the ideal phase accumulation ⟨q|U^(0)(T)|q⟩ = e^(iξq) connects the perturbative evolution to the gate phases.

C The C2Z Gate in the Linear Arrangement

In the linear arrangement, unequal blockade strengths constrain corrections to infinite-blockade C2Z pulses. Because the relevant computational states respond differently to finite blockade but identically to global-pulse changes, a small pulse adjustment cannot generally restore the gate.

Geometry and blockade strengths: The linear arrangement has two blockade strengths: B12 = B23 = B and B13 = B′ < B.In the van der Waals regime, B′ = B/64.
Finite-blockade correction: An infinite-blockade global pulse implementing C2Z cannot be corrected by a small change when B, B′ ≫ Ωmax.The statement applies to pulses such as Pulse 1 or Pulse 2 from Sec. 3.3.
Perturbative treatment: The perturbation from finite blockade can be treated as an ac-Stark shift in the limit B, B′ ≫ Ωmax.The perturbation also includes the change in the laser pulse.
Failure of small global corrections: C2Z requires equal phases for |011⟩ and |101⟩, but finite blockade makes their evolved amplitudes unequal when B ≠ B′.The two states behave identically under a global pulse, while their Hamiltonians are affected differently by the unequal blockade strengths.

D Approximation of a Finite Blockade Strength through an AC-Stark Shift

A time-dependent Schrieffer-Wolff transformation shows that finite blockade can be represented to first order by an ac-Stark shift, provided the pulse varies slowly enough relative to the blockade scale.

Transformation: The transformation removes couplings between |rr⟩ and the |W⟩ or |11⟩ states from the transformed Hamiltonian.It uses a time-dependent basis transformation e^−S(t).
Perturbative expansion: The generator and transformed Hamiltonian are expanded in powers of B^−1, with no S0 term because the B = ∞ Hamiltonian already has the relevant decoupling.The expansion is S = S1 + S2 + ... and ˜H11 = ˜H(−1)11 + ˜H(1)11 + ....
Approximation: To first order in 1/B, finite blockade is captured by an ac-Stark shift of |W⟩ when the transformed state is considered.At the initial and final times, the transformed and original states agree up to O(B^−2).
Validity condition: Neglecting higher-order terms requires both B ≫ |Ω| and B^2 ≫ |dotΩ|.The second condition controls terms involving the pulse derivative.
Validity condition: If Ω oscillates at a frequency of order B, finite-blockade effects cannot be described solely by an ac-Stark shift.In that regime, the required hierarchy in the perturbative expansion fails.

E Gate Error of a CZ and C2Z Gate to Second Order in 1/B

The paper derives perturbative finite-blockade errors for CZ and C2Z gates and compares them with exact calculations. The approximation agrees excellently for sufficiently large blockade strengths, while its validity depends on the pulse regime.

Perturbative error expansion: The appendix expands the gate states and fidelities in powers of 1/B for pulses that are perfect at B = ∞.The resulting expressions target CZ and triangular-arrangement C2Z gates.
Derivation: The CZ and C2Z error formulas are obtained by combining expansions of quantities and normalized states that depend on B.The derivation inserts the state expansions into the corresponding fidelity expressions.
Numerical comparison: Figure 9 compares exact finite-B gate errors with errors from Eqs. (54) and (55) using an ac-Stark-shift approximation.It covers the time-optimal CZ pulse and Pulses 1 and 2 for C2Z.
Numerical comparison: For all three pulses, the exact and approximate gate errors agree excellently for large B.The comparison is shown as blue solid and orange dotted or dashed curves in Fig. 9.