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Quantum Information Scrambling in a Superconducting Qutrit Processor
M. S. Blok, V. V. Ramasesh, T. Schuster, K. O'Brien, J. M. Kreikebaum, D. Dahlen, A. Morvan, B. Yoshida, N. Y. Yao, I. Siddiqi
TL;DR
Existing scrambling experiments used qubits, leaving higher-dimensional scrambling experimentally untested. This work builds a superconducting qutrit processor, implements scrambling operations, and embeds them in five-qutrit teleportation, obtaining Favg = 0.568 ± 0.001 and verifying scrambling.
Problem
Experimental scrambling demonstrations had used qubits, while higher-dimensional systems may exhibit distinct scrambling phenomena.
Method
The authors construct a two-qutrit scrambling unitary from controlled-SUM gates, characterize it with process tomography, and test it through five-qutrit teleportation.
Results
Favg = 0.568 ± 0.001, and the measured teleportation behavior verifies scrambling and upper-bounds the averaged OTOC at 0.618 ± 0.004.
Takeaways & Limitations
The experiment demonstrates a superconducting five-qutrit processor capable of coherently simulating multi-qutrit scrambling dynamics.
Abstract
from arXiv · showhide
The theory of quantum information provides a common language which links disciplines ranging from cosmology to condensed-matter physics. For example, the delocalization of quantum information in strongly-interacting many-body systems, known as quantum information scrambling, has recently begun to unite our understanding of black hole dynamics, transport in exotic non-Fermi liquids, and many-body analogs of quantum chaos. To date, verified experimental implementations of scrambling have dealt only with systems comprised of two-level qubits. Higher-dimensional quantum systems, however, may exhibit different scrambling modalities and are predicted to saturate conjectured speed limits on the rate of quantum information scrambling. We take the first steps toward accessing such phenomena, by realizing a quantum processor based on superconducting qutrits (three-level quantum systems). We implement two-qutrit scrambling operations and embed them in a five-qutrit teleportation algorithm to directly measure the associated out of-time-ordered correlation functions. Measured teleportation fidelities, Favg = 0.568 +- 0001, confirm the occurrence of scrambling even in the presence of experimental imperfections. Our teleportation algorithm, which connects to recent proposals for studying traversable wormholes in the laboratory, demonstrates how quantum information processing technology based on higher dimensional systems can exploit a larger and more connected state space to achieve the resource efficient encoding of complex quantum circuits.
I. INTRODUCTION
Qutrit processors offer a higher-dimensional platform for quantum information tasks, and this work demonstrates deterministic two-qutrit scrambling verified through a five-qutrit teleportation algorithm.
- Qutrit systems have been proposed to reduce code sizes, improve magic-state distillation, and strengthen quantum cryptography protocols.
- The work develops a superconducting multi-qutrit processor, implements a maximally scrambling two-qutrit unitary, and verifies it using five-qutrit teleportation.
- Quantum scrambling scatters initially localized information across available degrees of freedom while generating highly entangled states that enable thermalization in isolated quantum systems.
- Quantum processors can measure scrambling through the decay of out-of-time-ordered correlation functions, including with teleportation protocols resilient to decoherence and imperfections.
- The processor opens experimental access to quantum information processing based on qutrit logic.
II. QUTRIT PROCESSOR
Qutrits encode more than two states per site and can sometimes reduce system size and entangling-gate requirements, but deterministic universal two-qutrit gates had not previously been demonstrated.
- Qudits with d > 2 can store exponentially more information than qubits and may implement algorithms with fewer sites and multi-site entangling gates.
- Qutrit protocols have been proposed to provide advantages over qubit-based quantum information processing.
- Qutrits have been realized in photonic, solid-state, trapped-atom, trapped-ion, and superconducting systems.
- Before this work, no platform had demonstrated a deterministic, universal two-qutrit gate.
A. High-fidelity single qutrit operations
The processor addresses relaxation, dephasing, and cross-talk to enable fast, high-fidelity single-qutrit gates, while benchmarking leaves some qutrit-specific errors unresolved.
- A. High-fidelity single qutrit operations: Including the |2⟩ state introduces relaxation, charge-noise dephasing, and frequency-crowding cross-talk that must be mitigated for high-fidelity operations.
- A. High-fidelity single qutrit operations: T2,echo = (61.2 ± 11) µs and (28 ± 5) µs are achieved for the |0⟩→|1⟩ and |1⟩→|2⟩ transitions, respectively.The corresponding Ramsey dephasing times are T2* = (39±21) µs and (14±5) µs.
- A. High-fidelity single qutrit operations: Cross-talk is compensated by measuring the frequency-dependent matrix C(ω) and inverting it to correct drive-line fields.The calibration uses simultaneous drives, Stark shifts, or Rabi oscillations across relevant transition frequencies.
- A. High-fidelity single qutrit operations: f01 = 0.9997 ± 0.0001 and f12 = 0.9994 ± 0.0001 are the measured fidelities for gates in the {|0⟩, |1⟩} and {|1⟩, |2⟩} subspaces.Single-qutrit gates are performed within 30 ns.
- A. High-fidelity single qutrit operations: The benchmarking method is insensitive to idle-state phase errors and multi-qutrit errors, motivating genuine qutrit randomized benchmarking.
B. Two-qutrit entanglement
The processor uses cross-resonance and cross-Kerr interactions to implement two distinct neighboring-qutrit entangling gates. The conditional-π gate generates high-fidelity EPR pairs, while the controlled-SUM gate enables more general two-qutrit operations but is slower and less faithful.
- Entangling-gate architecture: The processor implements cross-resonance and cross-Kerr interactions as two methods for generating controllable entangling gates between neighboring qutrits.The cross-resonance interaction produces a conditional-π gate, while the cross-Kerr interaction is used for controlled-phase and controlled-SUM gates.
- Cross-resonance gate: The cross-resonance gate swaps the target’s |0⟩ and |1⟩ populations conditional on the control being |1⟩.Its dynamics also include an off-resonant effect on the target’s |2⟩ state and control-state-dependent Rabi oscillations.
- Cross-resonance gate: ω0−ω1 = 4 MHz and tg = 125 ns define the demonstrated conditional-π cross-resonance gate.The drive power is selected so ω0 = ω2, and the gate time satisfies tg|ω0 −ω1| = π.
- EPR preparation: FEPR = 0.98 ± 0.002 is achieved after two applications of the conditional-π gate, with fidelity mostly limited by decoherence.The EPR state is reconstructed using qutrit state tomography.
- Gate scope and trade-offs: The conditional-π gate acts only in a two-qubit subspace, so general two-qutrit unitaries require many such gates.The controlled-SUM gate is introduced to reduce this burden and is used for the complex two-qutrit scrambling operation.
- Controlled-SUM gate: The controlled-SUM gate reaches fidelity 0.889, with implementation time ∼1.5 µs per specified qutrit pair and decoherence as the primary limitation.The gate is constructed from the generic cross-Kerr interaction and single-qutrit rotations, with robustness to variability in interaction coefficients.
III. QUTRIT SCRAMBLING
This section presents a two-qutrit Clifford unitary designed to maximally scramble information, extending experimental scrambling studies beyond qubits. The unitary delocalizes every non-identity single-qutrit Pauli operator across both qutrits.
- Motivation: Experimental scrambling studies had previously used qubits, while higher-dimensional systems may exhibit distinct scrambling phenomena.The paper identifies qutrit scrambling as a step toward studying high-dimensional composite quantum systems.
- Scrambling unitary: The implemented unitary Us is a simple Clifford scrambling operation that permutes the nine two-qutrit computational basis states.Its action on Pauli operators remains within the Pauli group, establishing its Clifford character.
- Scrambling unitary: Us transforms each single-qutrit Pauli operator into a two-qutrit operator, satisfying the defining requirement of scrambling.This transformation demonstrates complete delocalization of initially local operators.
- Motivation: A two-qutrit system is the smallest bipartite system that permits maximal scrambling.The paper contrasts this with two-qubit systems, where complete delocalization of all single-qubit operators is impossible.
- Verification: The implementation is verified through quantum process tomography and a five-qutrit teleportation protocol.These two approaches provide explicit process characterization and teleportation-based verification of maximal scrambling.
A. Verifying scrambling through quantum process tomography
Quantum process tomography characterizes the implemented scrambling unitary and visualizes how it transforms local operators. The measured process confirms maximal scrambling while revealing hardware errors during cross-Kerr evolution.
- Implementation and tomography: The scrambling unitary Us is built from two sequential controlled-SUM gates with exchanged control and target qutrits.Full tomography reconstructs a 9 × 9 output density matrix for each of 81 input states, requiring millions of measurements.
- Implementation and tomography: Approximately 1 hour is sufficient to measure the full process matrix on superconducting circuits operating at repetition rates up to ∼100 kHz.The passage contrasts this with platforms operating at duty cycles in the Hz range, where the procedure would be prohibitively long.
- Experimental result: 0.875 is the fidelity of the implemented scrambling operation measured by quantum process tomography.The dominant error mechanisms are dephasing and amplitude-damping during cross-Kerr evolution.
- Experimental result: Process tomography verifies that Us maps all single-qutrit operators to fully two-qutrit operators and maps Pauli operators to different Pauli operators.The controlled-SUM comparison is entangling but leaves Z† ⊗I and I ⊗X† undelocalized.
B. Verifying scrambling through quantum teleportation
The five-qutrit teleportation protocol tests scrambling by requiring the input state to reach the output after EPR measurements, while distinguishing scrambling from decoherence. Using a maximally scrambling unitary yielded teleportation above the qutrit classical limit despite experimental imperfections.
- Motivation: Process tomography scales exponentially with system size, whereas teleportation provides a more scalable but essentially one-parameter scrambling diagnostic.The teleportation protocol quantifies how operators on one qutrit spread onto another through averaged OTOCs.
- Protocol: The protocol applies Us and its conjugate U*s to separate qutrit pairs initially linked by EPR entanglement, then measures Q2 and Q3 in the EPR basis.Conditioned on the EPR outcome, the input state in Q1 can be teleported to Q5 when the dynamics scramble information.
- Implementation: The experiment realizes EPR preparation, Us and U*s, reverse EPR decoding for computational-basis readout, and qutrit dynamical decoupling against always-on cross-Kerr interactions.The EPR measurement is heralded by outcomes in which Q2 and Q3 are measured in |00⟩.
- Control: The identity control, implemented with the same gate count and types as the scrambler, produces Fψ ≈ 1/3 and a nearly maximally mixed Q5 state.This control separates the scrambling signal from errors associated with circuit complexity.
- Measurement: Teleportation fidelity above 1/2 verifies non-zero qutrit scrambling, while the sampled twelve-state ensemble gives an unbiased average over all single-qutrit pure states.The twelve states form a state 2-design, and the average fidelity can upper-bound the associated averaged OTOCs without assuming a noise model.
- Results: Favg = 0.568 ± 0.001 for the maximally scrambling unitary, with all but one input state above 1/2, and the averaged OTOC is upper-bounded by 0.618 ± 0.004.The result verifies coherent multi-qutrit scrambling despite experimental error.
IV. CONCLUSION
The work demonstrates a five-qutrit superconducting processor with entangling gates, tomography, and dynamical decoupling sufficient for verifying qutrit scrambling. It also identifies qutrit error correction and broader ternary-logic applications as extensions.
- Conclusion: The experiment demonstrates a five-qutrit processor built from superconducting transmon circuits.
- Conclusion: High-fidelity single- and two-qutrit gates, tomography, and dynamical decoupling support benchmarking, debugging, and simultaneous operation on adjacent qutrits.
- Conclusion: The teleportation protocol is equivalent to a three-qutrit error-correcting code that protects information from erasure of any one of the three qutrits.
- Future directions: Future directions include more efficient ternary Toffoli decompositions, improved magic-state distillation, broader spin-1 platforms, and higher-dimensional qudit processors.
Appendix A: Processor and fabrication details
The processor uses five fixed-frequency transmon qutrits integrated into an eight-transmon ring, with niobium circuitry, aluminum junctions, multiplexed readout, coupling resonators, and microwave control.
- Chip architecture: Five fixed-frequency single-junction transmon qutrits occupy a chip with an eight-transmon ring geometry.
- Materials: Niobium forms the resonators, Purcell filter, capacitors, drive lines, and ground plane, while aluminum with an aluminum-oxide barrier forms the transmon junctions.
- Connectivity: Each transmon couples to a linear readout resonator, two nearest-neighbor coupling resonators, and a microwave drive line.
- Readout: Readout resonators have effective linewidth κext ≈ 1 MHz and share a common λ/2 Purcell filter with external Q ≈ 10.
- Packaging: Wirebonds suppress slotline modes and allow the readout bus to overlap a coupling resonator.
Appendix B: Experimental setup
The experimental setup characterizes qutrit lifetimes, dephasing, control errors, cross-Kerr interactions, and mediated exchange couplings using cryogenic control and measurement hardware.
- Cryogenic setup: The chip is operated at 10 mK with separate local oscillators for qubit control, readout, and TWPA pumping.
- Readout chain: Readout signals are amplified by a TWPA at 10 mK, HEMT amplifiers at 4 K, and room-temperature electronics before digitization at 1.25 GSa/s.
- Characterization: Lifetimes and dephasing times are extracted by fitting decay curves, while randomized benchmarking measures single-qutrit pulse errors across subspaces.
- Interactions: Neighboring qutrits’ cross-Kerr coefficients are measured with Ramsey experiments using |1⟩ and |2⟩ states, and non-nearest-neighbor residuals are negligible.
- Interactions: The cross-Kerr interaction is the dispersive limit of nearest-neighbor exchange mediated by coupling resonators, with measured g roughly 3 MHz on a comparable tunable chip.
2. Coherence of third transmon level
The processor’s third transmon level introduces faster damping and dephasing, requiring design choices and control methods to preserve qutrit coherence. Crosstalk compensation enables selective driving but is measurement-intensive and difficult to scale.
- The |2⟩ state decays to |1⟩ roughly twice as fast as |1⟩ decays to |0⟩.
- Dephasing from charge noise occurs roughly an order of magnitude faster for each successive transmon level, particularly |2⟩ relative to |1⟩.
- Increasing EJ/EC from approximately 50 to 73 reduced charge dispersions but also lowered anharmonicity from roughly 300 MHz to 250 MHz.
- Crosstalk: Crosstalk is modeled by a frequency-dependent five-by-five complex matrix relating input and output microwave fields, enabling frequency-specific cancellation by matrix inversion.
- Crosstalk: The cancellation method requires many measurements and failed to compensate some approximately 10 ns pulses, limiting scalability to larger processors.
- Two-qutrit control: When neighboring qutrits occupy superpositions, an ordinary controlled-phase implementation can entangle them through shared cross-Kerr coupling.
Appendix D: Dynamically-decoupled EPR preparation
Simultaneous EPR-pair preparation creates unwanted cross-Kerr-mediated entanglement between neighboring qutrits. Dynamical decoupling uses tailored permutation pulses to suppress this interaction while preserving the desired gate operations.
- Six free-evolution periods interleaved with local permutation pulses implement a diagonal phase gate while protecting qutrits from dephasing and static-neighbor interactions.
- For parallel gates, swapping permutation order between qutrit pairs decouples their shared neighbor while retaining the desired operations.
- Simultaneous EPR preparation lowers pair coherence because cross-Kerr coupling creates unwanted entanglement between Q3 and Q4.
- Tomography shows that most unwanted entanglement appears after Q3 becomes populated in the |2⟩ state.
- The sequence therefore applies dynamical decoupling only after |2⟩ population begins, avoiding extra single-qutrit-gate errors during initial Bell-state preparation.
a: Individual preparation
The figure distinguishes individual and simultaneous EPR-pair preparation, including a dynamically decoupled simultaneous sequence. Individual preparation yields substantially higher state fidelities than simultaneous preparation without decoupling.
- c: Simultaneous preparation with dynamical decoupling: The third condition is simultaneous preparation with dynamical decoupling, completing the figure’s comparison of preparation strategies.
- a: Individual preparation: Individual EPR-pair preparation achieves state fidelities of 0.94±0.002 for Q2/Q3 and 0.98±0.002 for Q4/Q5.
- b: Simultaneous preparation: Simultaneous preparation without dynamical decoupling reduces the corresponding fidelities to 0.81±0.002 and 0.82±0.002.