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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
TL;DR
Parallel entangling gates in fully connected ion systems are difficult because shared motional modes can produce crosstalk. The paper develops optimized optical pulse controls for simultaneous two-qubit gates and demonstrates their use in trapped-ion experiments, including a four-qubit GHZ-state proposal. The experiments obtain typical fidelities of 96–99%, while parallel operation requires more optical power and the GHZ extension needs new calibration strategies.
Problem
Shared common-mode couplings complicate parallel entangling gates through potential crosstalk, limiting experimental evidence for such operations in fully connected trapped-ion systems.
Method
The paper uses nonlinear optimization of independently shaped optical pulses to implement simultaneous entangling operations while controlling motional closure and unwanted interactions.
Results
Typical parallel-gate fidelities are 96–99%, with reported examples of 96.4(3)% and 99.4(3)% and average crosstalk error 2.2(3)%.
Takeaways & Limitations
The control scheme supports parallel two-qubit operations and suggests reduced-depth multi-qubit entanglement, including a possible single-operation GHZ state.
Takeaways & Limitations
Parallel gates require more optical power, while the proposed GHZ extension may need independent pulses and new calibration techniques because six interactions must be controlled with four signals.
Abstract
from arXiv · showhide
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
The method designs parallel xx gates by constraining spin-motion closure, desired pairwise entanglement, and zero crosstalk, then optimizing segmented pulse amplitudes.
- 4N spin-motion terms must vanish at the gate end so all motional trajectories close and residual qubit-motion entanglement is removed.
- Only the two target ion pairs receive χideal entanglement, while the four crosstalk pairs are constrained to zero interaction.χideal = π/4 implements a maximally entangling xx gate; smaller positive values produce partial entanglement.
- The resulting optimization controls 4N spin-motion parameters and six spin-spin interaction parameters for two simultaneous pairs.The spin-motion constraints are linear, whereas the six spin-spin constraints are nonlinear.
- Distinct piecewise-constant pulse signals are assigned to each desired ion pair, with amplitudes varied independently across equal-duration segments.Each ion’s pulse is divided into S segments of duration τ/S.
- The displacement and spin-spin interactions are expressed through segment-dependent control matrices constructed from motional frequencies, detuning, couplings, and time ordering.The segment formulation enforces s ≤ s′ when evaluating the time-ordered double integral.
OPTIMIZATION METHODS
Pulse design is converted into an unconstrained numerical optimization that balances deviations from ideal gate conditions against optical-power cost.
- The penalty objective minimizes quadratic deviations of α and χ from their ideal values while penalizing high-power pulse sequences.The optimization uses MATLAB’s built-in fminunc function.
FIDELITY OF PARALLEL xx OPERATIONS
The fidelity analysis evaluates simultaneous xx operations from residual motion, interaction-angle errors, crosstalk, and thermal phonon occupations. Ideal closure and interaction parameters give unit fidelity.
- The simultaneous-gate fidelity depends on crosstalk combinations involving interactions between each target pair and the ions in the other pair.These terms include χim, χin, χjm, and χjn, together with Δχij and Δχmn.
- The fidelity model also includes multi-ion interaction combinations that couple deviations across both target gates and their crosstalk pairs.
- F|| = 1 for ideal closed trajectories, zero crosstalk interactions, and target interaction strengths.The ideal-case calculation sets α = 0, crosstalk χ terms to zero, and target χ values to χideal.
TOWARD A SINGLE-OPERATION GHZ STATE
The control strategy suggests extending parallel pairwise gates to single-operation four-qubit GHZ-state generation. This could reduce circuit depth, but the extension requires more independent controls and new calibration methods.
- Setting all six spin-spin interactions to π/4 while closing all motional trajectories produces a four-qubit GHZ state.
- The proposed GHZ fidelity is evaluated from deviations of all six interaction strengths from their ideal values.
- The four-ion GHZ extension may require independent pulse shapes on all ions because six equal-strength interactions must be controlled with only four signals.Calibration may need additional pulse-section power controls rather than one overall scaling factor.
- The proposed GHZ operation remains an indication of feasibility, with experimental calibration challenges and the need for more degrees of freedom.
- Parallel pairwise gates reduce GHZ construction depth from O(N) sequential two-qubit gates to O(log(N)) using a binary-tree schedule.A single-operation GHZ state would reduce the depth to unity.
EXPERIMENTAL SETUP
The experiments use a five-ion 171Yb+ chain with hyperfine clock-state qubits, Raman-beam control, and independent addressing electronics. Qubits are optically initialized and read through state-dependent fluorescence.
- The experiment uses a linear chain of five trapped 171Yb+ ions cooled near their motional ground state.
- Qubit states are the magnetic-field-insensitive hyperfine states |0⟩ and |1⟩, separated by 12.642821 GHz.
- Counterpropagating Raman beams and a multi-channel AWG provide individual phase, frequency, and amplitude control for parallel two-qubit operations.
- Optical pumping initializes qubits to |0⟩, while a multi-channel photomultiplier array performs state-dependent fluorescence readout.
CALCULATING FIDELITIES OF 2-QUBIT ENTANGLING GATES
Gate fidelity is estimated by parity analysis: measured parity after a phase-scanned analysis rotation supplies the parity amplitude used with populations in the fidelity expression. The procedure compares the experimental density matrix with the ideal xx(χ) operation.
- A phase-scanned global π/2 analysis rotation is applied after the xx gate, and parity is measured across the scan.
- Parity analysis covers all 6 ion pairs among the 4 ions, including 2 intended entangled pairs and 4 crosstalk pairs.
- The fidelity compares the experimentally produced general density matrix ρg with the ideal fidelity matrix ρideal.
- Parity is the sum of even-parity populations minus the sum of odd-parity populations, with coherences represented through amplitudes and phases.
- The parity scan fits a sine curve to estimate AΠ, which is inserted into the xx(χ) fidelity formula alongside measured |00⟩ and |11⟩ populations.
ADDITIONAL PARITY CURVES AND FIDELITY DATA FOR 2-QUBIT ENTANGLING GATES
Additional parity curves show that parallel entangling gates generally achieve 96–99% fidelity, with one lower-fidelity pair attributed to a non-ideal pulse solution.
- 96–99% is the typical fidelity range for the additional parallel entangling gates.
- The {(1,2), (4,5)} configuration is an exception, with the (4,5) gate reaching 91% fidelity.
- The low (4,5) fidelity is attributed to a non-ideal pulse solution shown by its phase-space closure diagram.
FIDELITY OF PARALLEL 2-QUBIT ENTANGLING GATES WITH DIFFERENT DEGREES OF ENTANGLEMENT
The parallel-gate scheme independently calibrates the entanglement applied to each ion pair by scaling each gate’s optical power. Experiments demonstrate unequal entanglement strengths with high pair fidelities and measurable crosstalk.
- Independent calibration makes the χ parameters of the two parallel xx gates independently adjustable.
- Scaling the overall gate power changes χ, allowing two simultaneous xx gates to generate different degrees of entanglement.
- For ions (1,5) and (2,4), χ = π/4 and χ = π/8 produce fidelities of 96.4(3)% and 99.4(3)%, respectively, with 2.2(3)% average crosstalk error.
- Across several ion-pair configurations, parallel xx gates yield pair fidelities from 91.9(3)% to 98.4(3)% and average crosstalk errors from 0.6(3)% to 1.7(3)%.
INDEPENDENCE OF PARALLEL GATE CALIBRATION
Parallel gates can be calibrated independently by adjusting optical-power scaling factors without changing pulse shapes or degrading the other gate. Population scans and parity measurements show high fidelities with low crosstalk.
- Independent calibration: Adjusting one ion’s scaling factor changes the entanglement of its participating gate without apparent ill effects on the other gate.Scans with the (3,4) interaction on show that populations for that gate remain unchanged while the (1,2) gate is adjusted.
- Gate performance: 96.4(3)% and 99.4(3)% fidelities were obtained for the two parallel xx gates, with 2.2(3)% average crosstalk error.The parity curve compares simultaneous xx(χ) gates on ion pairs (1,5) and (2,4).
- Calibration resources: Table I compares each component parallel xx gate’s optical power with its corresponding stand-alone two-qubit gate using the ratio R||.The comparison is organized by each experimentally implemented pair of parallel gates.
OPTICAL POWER REQUIREMENTS
Parallel execution reduces the time needed for two entangling gates but requires additional optical power. Most tested parallel gates used roughly two to four times the power of corresponding standalone gates.
- Timing: 250 µs is comparable to one xx-gate time and half the duration of executing two xx gates serially.This timing applies to running two xx gates in parallel.
- Power metric: The parallel-to-standalone power comparison is expressed through R||, with beam areas canceling because beam sizes do not vary.Intensity is power per unit area, so fixed beam areas permit the power-ratio calculation.
- Power cost: Most parallel gates require about two to four times as much optical power as their singly performed counterparts.Some solutions required substantially more power, including difficulty finding a high-quality, low-power solution for pairs (1,2) and (3,4).
OPTIMIZED ADDER CIRCUIT
The optimized full-adder circuit combines and further optimizes component gates, with two parallel two-qubit operations explicitly incorporated. Calibration scans distinguish operation with the companion gate on or off.
- Circuit construction: The optimized full-adder circuit combines cnot, C(V), and C(V†) gates and further optimizes its rotations.The construction is based on an earlier circuit and places the two parallel two-qubit operations in dashed boxes.
- Calibration verification: Figure 7 scans ion-1, ion-2, or joint (1,2) scaling factors while the (3,4) gate is either active or inactive.Panels (a) and (b) keep the (3,4) gate on; panels (c) and (d) set its scale factor to zero.
- Parallel operations: Figure 8 identifies two parallel 2-qubit operations within the application-optimized full-adder implementation.The parallel operations are outlined with dashed boxes.