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Parallel Entangling Operations on a Universal Ion Trap Quantum Computer

C. Figgatt, A. Ostrander, N. M. Linke, K. A. Landsman, D. Zhu, D. Maslov, C. Monroe

arXiv:1810.11948v1quant-phcs.ET

TL;DR

Parallel entangling gates can speed quantum computations, including full adders, but require control of simultaneous operations. The paper uses numerical optimization to implement parallel entangling gates in trapped ions and demonstrates a full-adder computation with an average process fidelity of 83.3(3)% .

  • Problem

    Parallel entangling operations could speed quantum computation, including full adders, but their implementation in fully connected systems remains challenging.

    Source: pp. 2, 4

  • Method

    The authors use numerical optimization to find pulse solutions for simultaneous entangling operations in trapped-ion qubits.

    Source: pp. 2, 4

  • Results

    83.3(3)% average process fidelity was achieved for the demonstrated quantum full-adder computation.

    Source: p. 4

  • Takeaways & Limitations

    Parallel 2-qubit operations provide a tool to speed computation while avoiding spectral crowding as processor size grows.

    Source: p. 4

    Higher theoretical solution fidelity may require easing power constraints, although greater experimental power can exacerbate Raman beam noise.

    Source: p. 4

Abstract

from arXiv · show

The circuit model of a quantum computer consists of sequences of gate operations between quantum bits (qubits), drawn from a universal family of discrete operations. The ability to execute parallel entangling quantum gates offers clear efficiency gains in numerous quantum circuits as well as for entire algorithms such as Shor's factoring algorithm and quantum simulations. In cases such as full adders and multiple-control Toffoli gates, parallelism can provide an exponential improvement in overall execution time. More importantly, quantum gate parallelism is essential for the practical fault-tolerant error correction of qubits that suffer from idle errors. The implementation of parallel quantum gates is complicated by potential crosstalk, especially between qubits fully connected by a common-mode bus, such as in Coulomb-coupled trapped atomic ions or cavity-coupled superconducting transmons. Here, we present the first experimental results for parallel 2-qubit entangling gates in an array of fully-connected trapped ion qubits. We demonstrate an application of this capability by performing a 1-bit full addition operation on a quantum computer using a depth-4 quantum circuit. These results exploit the power of highly connected qubit systems through classical control techniques, and provide an advance toward speeding up quantum circuits and achieving fault tolerance with trapped ion quantum computers.

METHODS · CONSTRAINT PROBLEM FOR OPTIMAL PARALLEL OPERATIONS

For parallel xx gates on two ion pairs, the method formulates constraints that close spin-motion trajectories, entangle only the intended pairs, and suppress crosstalk. It optimizes separate pulse shapes using piecewise-constant amplitudes and interaction constraints.

CONSTRAINT PROBLEM FOR OPTIMAL PARALLEL OPERATIONS

  • The gate requires all 4N spin-motion terms to vanish, while the two target pairs acquire χideal = π/4 and the four crosstalk pairs acquire zero interaction.Smaller positive χ values implement partially entangling gates.Source: CONSTRAINT PROBLEM FOR OPTIMAL… · p. 6
  • This two-pair optimization has 4N + 6 parameters to control.The parameter count follows from the spin-motion closure and target-versus-crosstalk interaction requirements.Source: CONSTRAINT PROBLEM FOR OPTIMAL… · p. 6
  • The gate amplitude is divided into S equal-duration segments, with independently varied amplitudes and distinct pulse shapes for the two target pairs.Each pair receives its own signal: ions (i, j) share one pulse shape and ions (m, n) another.Source: CONSTRAINT PROBLEM FOR OPTIMAL… · p. 6
  • Precomputed interaction constants depend on the motional frequencies, detuning, and segment indices, and are arranged into matrices for the spin-motion and spin-spin terms.Time ordering of the spin-spin double integral requires s ≤ s′, including a constraint within segments when s = s′.Source: CONSTRAINT PROBLEM FOR OPTIMAL… · p. 7
  • Segment amplitudes for the two target pairs are arranged as separate vectors, with spin-motion and spin-spin interactions represented by corresponding matrices.The formulation also includes matrices for the four crosstalk-pair interactions.Source: CONSTRAINT PROBLEM FOR OPTIMAL… · p. 7
  • The resulting C-constraints are linear, whereas the six D-constraints on spin-spin interactions are nonlinear.These constraints encode the spin-motion and intended-versus-crosstalk interaction requirements.Source: CONSTRAINT PROBLEM FOR OPTIMAL… · p. 7

OPTIMIZATION METHODS

The optimization converted constraints on α and χ into an unconstrained objective that balanced deviations from ideal values against high pulse power, then minimized it with MATLAB’s fminunc.

OPTIMIZATION METHODS

  • The penalty-method objective was quadratic in α and χ deviations from their ideal values and penalized high-power pulse sequences.Source: OPTIMIZATION METHODS · p. 7
  • The objective function was minimized using MATLAB’s built-in fminunc function.Source: OPTIMIZATION METHODS · p. 7
  • The initial pulse-shape guess assigned one pair all-positive amplitudes and the other pair positive amplitudes in the first half and negative amplitudes in the second half.Source: OPTIMIZATION METHODS · p. 7

FIDELITY OF PARALLEL xx OPERATIONS

The section derives fidelity for simultaneous XX gates as a function of control parameters and shows that ideal-case parameters yield unit fidelity.

FIDELITY OF PARALLEL xx OPERATIONS

  • The derived fidelity expression is an expansion of Γ terms and cosine-weighted combinations of the phase parameters.The displayed terms combine Γ coefficients with cosine functions of sums and differences of Δχ and χ phase parameters.Source: FIDELITY OF PARALLEL xx OPERATIONS · pp. 7–8
  • The expression continues with additional Γ-weighted cosine terms involving Δχij, χim, χin, χjm, χjn, and Δχmn.These terms complete the displayed fidelity expression.Source: FIDELITY OF PARALLEL xx OPERATIONS · p. 8
  • F|| = 1 when the ideal-case parameters are used, with zero motion displacements and zero listed phase shifts.The ideal conditions set α_{i,j,m,n,k}(τ) and χ_im, χ_in, χ_jm, χ_jn to zero and χ_ij to χ_ideal.Source: FIDELITY OF PARALLEL xx OPERATIONS · p. 8

TOWARD A SINGLE-OPERATION GHZ STATE

The section proposes a single-operation route to four-qubit GHZ preparation using the parallel ion-gate control scheme, while outlining pulse-control and calibration challenges and potential circuit-depth gains.

TOWARD A SINGLE-OPERATION GHZ STATE

  • Setting all six spin-spin interactions to χ = π/4 in Equation 1 yields a unitary that produces a 4-qubit GHZ state.Source: TOWARD A SINGLE-OPERATION GHZ STATE · pp. 8–9
  • The objective fidelity expression combines coherence factors Γ with cosine terms in deviations Δχ across the six pairwise interactions.Source: TOWARD A SINGLE-OPERATION GHZ STATE · p. 9
  • Producing these states may require independent pulse shapes on all four ions and new calibration controls to equalize six interactions using only four control signals.Adjusting only the overall pulse power may no longer work; the passage suggests independently adjusting power in several pulse sections on each ion.Source: TOWARD A SINGLE-OPERATION GHZ STATE · p. 10
  • For N qubits, GHZ circuit depth drops from O(N) with 2-qubit gates to O(log(N)) with parallel gates, and to unity with a single-operation gate.The passage further says arbitrary stabilizer circuits can shift from O(N^2 log(N)) CNOT gates to O(N) gates with single-operation GHZ circuitry. Unlike axial-mode multi-qubit GHZ demonstrations, this scheme uses radial-mode interactions.Source: TOWARD A SINGLE-OPERATION GHZ STATE · p. 10

EXPERIMENTAL SETUP

The experiments used a laser-cooled chain of five 171Yb+ ions, with individually addressed Raman-beam control for parallel two-qubit operations and fluorescence readout.

EXPERIMENTAL SETUP

  • A linear chain of five laser-cooled 171Yb+ ions encoded qubits in magnetic-field-insensitive clock states split by 12.642821 GHz.The ions were cooled near their ground state.Source: EXPERIMENTAL SETUP · p. 10
  • Counterpropagating Raman beams from a 355 nm laser drove coherent operations, while individual addressing beams and AWG controls enabled parallel two-qubit operations.The AWG supplied separate RF signals controlling each ion’s phase, frequency, and amplitude.Source: EXPERIMENTAL SETUP · p. 10
  • Qubits were initialized to |0⟩ by optical pumping and read out through state-dependent fluorescence using separate channels of a multichannel PMT array.Source: EXPERIMENTAL SETUP · p. 10

CALCULATING FIDELITIES OF 2-QUBIT ENTANGLING GATES

The section describes estimating 2-qubit xx(χ) gate fidelity from parity measurements after a phase-scanned analysis rotation. A fitted parity amplitude and measured populations are used to evaluate fidelity against the ideal gate.

CALCULATING FIDELITIES OF 2-QUBIT ENTANGLING GATES

  • A phase scan of a global π/2 rotation after the xx gate yields parity measurements used to assess gate fidelity.The analysis phase φ is scanned and parity is calculated at each point.Source: CALCULATING FIDELITIES OF 2-QUBIT… · p. 10
  • SK1 composite analysis pulses improve robustness to rotation-angle errors, and all six pairs among four ions are analyzed for entangled-pair and crosstalk behavior.The six pairs comprise two entangled ion pairs and four crosstalk pairs.Source: CALCULATING FIDELITIES OF 2-QUBIT… · p. 10
  • Fidelity is calculated from the measured |00⟩ and |11⟩ populations and parity amplitude AΠ, extracted by fitting a sine curve to the phase-scanned parity.For an ideal maximally entangling xx(π/4) gate, the parity curve has amplitude 1 and two periods over 0 to 2π.Source: CALCULATING FIDELITIES OF 2-QUBIT… · p. 11

ADDITIONAL PARITY CURVES AND FIDELITY DATA FOR 2-QUBIT ENTANGLING GATES

Additional parity curves show typical 2-qubit entangling-gate fidelities of 96–99%, with a lower-fidelity exception attributed to a nonideal pulse solution.

ADDITIONAL PARITY CURVES AND FIDELITY DATA FOR 2-QUBIT ENTANGLING GATES

  • 96-99% typical fidelities were measured; the (4,5) gate in {(1,2), (4,5)} reached 91%, attributed to a nonideal pulse solution.The additional parity curves and corresponding fidelities are shown in Figure 5; the phase-space closure diagram in indicates the pulse solution is not ideal.Source: ADDITIONAL PARITY CURVES AND FIDELITY… · p. 11

FIDELITY OF PARALLEL 2-QUBIT ENTANGLING GATES WITH DIFFERENT DEGREES OF ENTANGLEMENT

Independent calibration lets the two parallel xx gates use different entanglement settings. In a demonstration, the gates achieved fidelities of 96.4(3)% and 99.4(3)%, with average crosstalk error of 2.2(3)%.

FIDELITY OF PARALLEL 2-QUBIT ENTANGLING GATES WITH DIFFERENT DEGREES OF ENTANGLEMENT

  • Because the xx gates are calibrated independently, their χ parameters can differ, and χ determines the amount of entanglement generated.Source: FIDELITY OF PARALLEL 2-QUBIT… · p. 11
  • Scaling the overall gate power adjusts χ, enabling simultaneous xx gates with different entanglement settings, as required by the full-adder implementation.Source: FIDELITY OF PARALLEL 2-QUBIT… · p. 11
  • 96.4(3)% and 99.4(3)% were the respective gate fidelities for simultaneous xx gates on ions (1,5) and (2,4), with average crosstalk error of 2.2(3)%.The (1,5) gate used χ = π/4, while the (2,4) gate used χ = π/8.Source: FIDELITY OF PARALLEL 2-QUBIT… · p. 11

INDEPENDENCE OF PARALLEL GATE CALIBRATION

Parallel gates can be calibrated independently by scaling power without changing pulse shape; experimentally, adjusting ions 1 and 2 changed their gate populations while leaving ions (3,4) unchanged when that gate was on or off.

INDEPENDENCE OF PARALLEL GATE CALIBRATION

  • Scaling gate power without changing pulse shape allows independent calibration, and scaling one ion affects only its participating gate by changing entanglement without apparent ill effects on gate quality.This behavior was experimentally confirmed during parallel operations on ions (1,2) and (3,4).Source: INDEPENDENCE OF PARALLEL GATE… · p. 12
  • Scaling ions 1 and 2 increased their |11⟩ population and decreased |00⟩, while the (3,4) populations remained unchanged with that gate on or off.The scans varied ion 1 alone, ion 2 alone, or ions 1 and 2 together; the (3,4) gate was set near fully entangling or to zero power.Source: INDEPENDENCE OF PARALLEL GATE… · pp. 12–13

OPTICAL POWER REQUIREMENTS

Parallel xx gates reduce execution time but require more optical power: most use about two to four times the power of singly performed gates, with some requiring substantially more.

OPTICAL POWER REQUIREMENTS

  • At τgate = 250 µs, two parallel xx gates take about the duration of one gate, half the time of two serial gates, but require more optical power.Source: OPTICAL POWER REQUIREMENTS · p. 13
  • Most parallel gates require about two to four times the optical power of singly performed counterparts, though some gate pairs require substantially more.The power estimate uses Ω ∝ √(I0I1); because beam sizes are fixed, intensity tracks power.Source: OPTICAL POWER REQUIREMENTS · p. 13
  • Since unused individual beams are dumped after the AOM, parallel operation trades twice or more the power for half the time versus serial operation.For two xx gates, the parallel scheme uses beams of power P to illuminate four ions for 250 µs; the serial scheme uses four beams at P/4 to P/2 for twice as long, wasting two beams.Source: OPTICAL POWER REQUIREMENTS · p. 13

OPTIMIZED ADDER CIRCUIT

The optimized full adder combines entangling gates with optimized rotations and includes two parallel 2-qubit operations.

OPTIMIZED ADDER CIRCUIT

  • The implemented adder combines CNOT, C(V), and C(V†) gates with optimized rotations in an xx(χ), rx(θ), and ry(θ) circuit containing two parallel 2-qubit operations.The circuit is constructed from Figure 4(b), with rotations optimized using the method described in Section 5.2.1 of.Source: OPTIMIZED ADDER CIRCUIT · p. 13, Figure 8 · p. 15
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