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Large Scale Modular Quantum Computer Architecture with Atomic Memory and Photonic Interconnects

C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L. -M. Duan, J. Kim

arXiv:1208.0391v2quant-ph

TL;DR

Scalable quantum computers need compatible quantum memories and buses, especially for operations between distant qubits. The paper analyzes a modular ion-trap architecture using local atomic interactions and probabilistic photonic links, showing fault-tolerant operation and feasibility for modest circuits, including quantified resource estimates for arithmetic tasks.

  • Problem

    Scalable quantum hardware must combine reliable quantum memories with controllable quantum buses, while existing QCCD approaches face complexity and long-distance communication limitations.

  • Method

    The paper analyzes MUSIQC, combining local trapped-ion registers with probabilistic photonic entanglement between registers and fault-tolerant constructions based on hypercells and cluster states.

  • Results

    The architecture is shown to support fault-tolerant computation; with two levels of concatenated Steane code, a 128-bit integer can be factored in less than 10 hours using fewer than 6 × 10^6 physical qubits.

  • Takeaways & Limitations

    MUSIQC provides a modular route to large-scale ion-trap processors in which photonic links connect registers and support efficient nonlocal arithmetic circuits.

Abstract

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The practical construction of scalable quantum computer hardware capable of executing non-trivial quantum algorithms will require the juxtaposition of different types of quantum systems. We analyze a modular ion trap quantum computer architecture with a hierarchy of interactions that can scale to very large numbers of qubits. Local entangling quantum gates between qubit memories within a single register are accomplished using natural interactions between the qubits, and entanglement between separate registers is completed via a probabilistic photonic interface between qubits in different registers, even over large distances. We show that this architecture can be made fault-tolerant, and demonstrate its viability for fault-tolerant execution of modest size quantum circuits.

I. INTRODUCTION

The paper proposes a modular ion-trap architecture that combines local atomic-ion registers with photonic links between registers to address scalability and long-distance communication. It analyzes fault tolerance and practical scaling limits while using demonstrated component technologies.

  • Fault tolerance: The architecture is argued to be fault-tolerant over a wide range of system parameters, with its component technologies demonstrated in small-scale trapped-ion systems.The paper also discusses technological hurdles remaining for realizing the complete architecture.
  • Motivation: Photonic communication addresses a limitation of QCCD scaling, whose ion shuttling and trap complexity do not easily extend over large distances.The paper identifies trap design, optical diffraction, and hardware-control complexity as additional scaling constraints.
  • Architecture: The architecture combines atomic-ion registers for quantum memory and local gates with photonic interconnects for flexible long-distance links between registers.Each Elementary Logic Unit contains trapped-ion qubits coupled through collective motion, while communication qubits connect to photonic channels.
  • Scalability: The proposed MUSIQC system may scale to 10^6 qubits using stable multi-qubit registers, ion shuttling, and scalable photonic interconnects.The architecture is presented as using component technologies that have already been demonstrated.
  • Local operations: Within an ELU, laser-driven state-dependent forces map qubit states onto collective phonon modes, enabling entangling operations between ion qubits.The characteristic gate speed is described as R_gate = ηΩ, with η the Lamb-Dicke parameter and Ω the motion-independent Rabi frequency.
  • Practical limits: Practical ELU size is constrained by crosstalk, motional heating, fluctuating fields, and slower gates as N_q grows; the paper estimates N_q = 10−100 as feasible.Long chains may require different-species refrigerator ions for sympathetic cooling, and extended ELUs can contain 20−1,000 physical qubits.

B. Probabilistic Linking of ELUs

MUSIQC links separate ion registers through probabilistic photonic entanglement between communication qubits. Type I connections favor higher success under low collection efficiency, while type II connections tolerate larger optical path fluctuations.

  • Photonic connections: Photonic interference between communication qubits generates entanglement that can mediate two-qubit gates between separate ELUs.The entangled communication qubits are used with local gates, measurements, and classical communication.
  • Type I interference: Type I connections weakly excite each communication qubit, and a single detected photon heralds entanglement with success probability p = peFηD.The relative optical path length must remain stable to much better than the optical wavelength.
  • Type II interference: Type II connections emit one photon from each communication qubit, and coincidence detection heralds entanglement with success probability p = (peFηD)2/2.The photon’s internal state, such as frequency, carries the qubit information.
  • Connection trade-offs: Type II connections are less sensitive to optical path fluctuations, requiring relative path stability only at the centimeter scale for typical hyperfine-encoded communication qubits.Their success probability may be lower than type I when light collection efficiency is low.
  • Connection times: For typical free-space collection, mean connection times are approximately 5 msec for type I and 250 msec for type II.The model assumes γ/2π = 20 MHz, F ∼10−2, and ηD ∼0.2; improved collection may enable times below 1 msec.
  • Practical constraints: Photonic entanglement requires isolating communication qubits so scattered excitation light and emitted photons do not disturb spectator memory qubits.Isolation may use shuttling or a different atomic species for communication qubits.

C. Reconfigurable Connection Network in MUSIQC

MUSIQC uses an optical crossconnect to flexibly pair many ELUs for remote entanglement generation. This network supports nested entanglement swapping, giving communication times that scale logarithmically with qubit separation under substantial parallelism.

  • Reconfigurable network: An OXC switch connects any available ELU input fiber to an output fiber feeding Bell-state detectors.Up to NELU/2 detectors can operate through the switch.
  • Hierarchical interactions: MUSIQC uses circuit-model computation within ELUs while generating inter-ELU Bell pairs probabilistically for remote operations.The architecture combines local gates with distributed entanglement resources.
  • Communication scaling: Nested entanglement swapping can reduce long-range communication from polynomial distance dependence to logarithmic scaling when many parallel operations are available.This addresses distant logical interactions required by complex n-bit algorithms.
  • Model assumptions: The fault-tolerant execution model assumes Steane [[7,1,3]] error correction with multiple concatenation levels and physical error levels near 10−7.The model estimates selected circuit execution times rather than providing a complete fault-tolerance analysis.

B. Construction of Efficient Arithmetic Circuits

The architecture targets efficient arithmetic by using QCLAs, whose logarithmic runtime requires long-range Toffoli operations supported by MUSIQC communication resources. The model estimates fault-tolerant execution using concatenated Steane encoding and parallel operations.

  • Arithmetic circuits: QCLA reduces adder runtime to O(log n), whereas nearest-neighbor QRCA runtime is O(n).QCLA requires additional qubits and parallel operations and outperforms QRCA for n above approximately 100.
  • MUSIQC implementation: MUSIQC flattens communication costs between ELUs, providing a platform for QCLAs with long-distance Toffoli interactions.QLA similarly uses a dedicated communication bus to reduce connection time.
  • Resource requirements: MUSIQC ELUs must support preparation of a logical |φ+⟩L state using at least three logical qubits, a 7-qubit cat state, and supporting ancillas.The design balances qubit resources against computation time using four ancillas per logical qubit.
  • Entanglement resources: At least three optical ports enable parallel entanglement generation for teleporting the gate operation, which requires seven entangled pairs per target ELU.Additional port and time-division multiplexity reduce entanglement-generation time.
  • Example hardware: The example MUSIQC ELU uses mp = 2 and mT = 10, requiring 100 qubits and 12 parallel operations per ELU.The selected multiplexities are intended to balance communication time with other operation times.
  • Fault-tolerant execution: The execution model applies error correction after each time step by measuring Steane-code stabilizers and performing the required corrections.Toffoli gates dominate the modeled circuit time.

D. QLA Implementation

QLA embeds logical qubits and communication units in a nearest-neighbor hardware layout while using dedicated communication resources for remote gates. Its Toffoli timing includes state preparation, entanglement distribution, and teleportation, with distribution depending logarithmically on distance.

  • QLA layout: A QLA logic unit contains 49 physical qubits hosting four logical qubits and ancilla qubits.Each logical qubit uses seven physical qubits and five ancillas within a 3 × 4 block.
  • Toffoli implementation: Fault-tolerant Toffoli execution prepares |φ+⟩L in an empty logic unit and teleports three qubits into that unit.The teleportation completes the gate operation.
  • Timing model: Toffoli execution time consists of |φ+⟩L preparation, entanglement distribution, and teleportation, with only distribution depending on LU distance.State preparation and teleportation are independent of distance in the model.
  • Distance scaling: QCLA Toffoli stages have qubit distances scaling as 2^t, with 1 ≤ t ≤ ⌊log2 n⌋.The corresponding communication-unit distance is analyzed in the two-dimensional QLA layout.
  • Entanglement distribution: Nested entanglement swapping requires approximately t/2 + 4 time steps for QCLA distance t.Each step includes one CNOT, two single-qubit gates, and one measurement.
  • Parallelism assumption: Achieving logarithmic communication time requires parallel two-qubit gates between every pair of qubits across the communication units.The n-bit adder model requires approximately 110n parallel operations.
  • Concatenation: At higher concatenation levels, QLA may omit explicit communication channels when distributed entanglement is sufficiently high quality to avoid purification and entanglement-pair error correction.This is described as an inter-level optimization.
  • Timing consequence: The logarithmic communication-time scaling enables effective estimation of gate-operation time with only small errors.

E. Results and Comparison

The resource and performance comparison evaluates fault-tolerant adders and modular exponentiation on MUSIQC, QLA, and nearest-neighbor architectures. MUSIQC uses more time than QLA for the example adder, but requires substantially fewer physical qubits for the modeled Shor computation.

  • Adder comparison: The comparison covers fault-tolerant QCLA on MUSIQC and QLA and QRCA on nearest-neighbor hardware.The circuits use one level of Steane [[7,1,3]] code in the Figure 5 comparison.
  • Adder comparison: QLA can execute QCLA faster than MUSIQC because its dedicated quantum bus enables remote gates with logarithmic distance dependence.The communication channel requires approximately three times as many physical qubits as the first encoded level’s stored and manipulated qubits.
  • Adder comparison: MUSIQC’s photonic network hampers execution time because establishing inter-register entanglement is probabilistic.The architecture dedicates substantial resources to speeding up entanglement generation.
  • Shor resource estimate: With 2 levels of concatenated Steane code, MUSIQC can factor a 128-bit integer in less than 10 hours using less than 6 × 10^6 physical qubits.The estimate concerns the modular exponentiation circuit, which represents running Shor’s algorithm.
  • Shor resource estimate: For the 128-bit case, QLA execution time is within 20% of MUSIQC, but its physical-qubit requirement is higher by about a factor of 10.The QLA single ELU exceeds 4.5 × 10^7 physical qubits, whereas MUSIQC uses approximately 58,000 ELUs with 100 qubits per ELU.

IV. FAULT TOLERANCE OF PROBABILISTIC PHOTONIC GATES

The analysis finds that MUSIQC can support scalable fault-tolerant quantum computation for any ratio of average entanglement creation time to decoherence time, even with additional gate errors. However, large ratios impose impractical qubit and time overheads.

  • Fault-tolerance result: Fault-tolerant quantum computation is possible for any ratio τE/τD, even in the presence of additional gate errors.The result establishes possibility rather than practical efficiency across all parameter ratios.
  • Fault-tolerance result: Large values of τE/τD lead to impractical overheads in qubits and time.The paper compares this overhead behavior to conventional fault tolerance near threshold error levels.

A. Analysis of fault-tolerance for fast entangling gates

For fast entangling gates, MUSIQC creates 3D cluster states from photonic Bell pairs, local CNOT gates, and measurements, then applies cluster-state fault-tolerance criteria. The approach has quantified error thresholds and constant operational architecture overhead, while requiring controlled error propagation and scheduling.

  • Cluster-state construction: MUSIQC creates a 3D cluster state through Bell-pair creation between ELUs, intra-ELU CNOT gates, and local measurements.The three-step procedure measures three of four qubits per ELU, with basis choices determined by whether the ELU represents a face or edge qubit.
  • Scheduling: The architecture adapts the sequence into five scheduled steps so qubits are never idle and are not acted on by multiple gates simultaneously.The latter scheduling constraint is required in some proposals for quantum gates with ion qubits.
  • Measurement scheduling: Non-Clifford-gate measurements require breaking 3D cluster states into overlapping slabs of bounded thickness to avoid measurement delays.Topological error correction and protected encoded Clifford gates require no adjustment of the measurement basis.
  • Error thresholds: The cluster-state fault-tolerance criterion has a 2.9% threshold error probability per memory step and measurement under the phenomenological error model.The cited criterion also covers gate-based and low-order correlated error models.
  • Error thresholds: The gate-based error model has a 0.67% threshold error probability per gate, with numerical tests agreeing well across varying local and two-local gate-error strengths.Error correction is performed using Edmonds’ perfect matching algorithm.
  • Operational overhead: Creating and locally measuring a 3D cluster state costs 54 gates per elementary cell in MUSIQC, compared with 24 gates in the standard setting.The resulting MUSIQC overhead over fault-tolerant cluster-state computation is constant.

B. Analysis of fault-tolerance for slow entangling gates

The hypercell architecture uses probabilistic photonic links and teleportation to connect modular ion-trap registers, remaining scalable even when entangling gates are slow. With finite gate errors, a second construction using nested 3D cluster states removes the distance-dependent error-growth limitation.

  • Hypercell Construction I: Arbitrarily large τE/τD ratios remain tolerable when ϵ = 0, because t can be reduced until memory error falls below the fault-tolerance threshold.The required operational cost increases with τE/τD but remains independent of computation size.
  • Hypercell Construction I: Hypercells use large probabilistic-link surface areas to create Bell pairs between neighboring structures, then teleport them to root qubits A and B.Each snowflake tree contains a root ELU and branching layers that provide many entanglement ports.
  • Hypercell Construction I: The probability that all m inter-hypercell attempts fail is Pfail = (1 −p)^m ≈ exp(−mp), while the root-to-root path length is l = 2 log2 m + 1.The number of available ports grows with the top layer of the tree, improving connection success.
  • Finite gate errors: With nonzero gate error ϵ, Construction I accumulates error along the entanglement-swapping path, imposing an upper bound on tree depth and on τE/τD.The resulting bound is necessary but not sufficient for fault-tolerant computation with these hypercells.
  • Hypercell Construction II: Construction II uses nested 3D cluster states so that, below the local error threshold, root-to-root Bell-pair error is independent of distance.Its total error is ϵtotal = c1t/τD + c2 ϵ, with constants independent of τE, τD, and root separation; sufficiently large inner clusters permit fault-tolerance for all τE/τD ratios with small gate errors.

V. OUTLOOK

The hierarchical modular ion-trap architecture is presented as scalable in both trapped-ion qubit count and the control structure required to manipulate those qubits. Its component technologies are described as available or within reach, supported by advances in surface-trap fabrication and control.

  • The architecture promises scalability in both the number of trapped-ion qubits and the hierarchical control structure used to manipulate them.
  • The technology required for every MUSIQC component is described as either already available or within reach.
  • Surface traps can be mapped onto two-dimensional surfaces fabricated with standard silicon microfabrication technologies.

Appendix A: Universal Fault-Tolerant QC using Steane Code

The appendix outlines fault-tolerant universal computation with the Steane [[7,1,3]] code, using transversal Clifford operations and a prepared logical state to implement Toffoli gates by teleportation. Long-distance CNOTs are performed by distributing entanglement rather than transporting the qubits themselves.

  • Logical-state preparation: Logical |0⟩L preparation measures six stabilizers with a reusable four-qubit cat state, repeating measurements up to three times for error correction.
  • Steane-code operations: The Steane [[7,1,3]] code supports transversal Pauli and Clifford operations, including logical CNOTs between nearby qubits.
  • Fault-tolerant Toffoli: Because Toffoli is non-Clifford and lacks a transversal Steane-code implementation, fault-tolerant execution prepares a special three-logical-qubit state and teleports the gate into it.
  • Remote gates: Long-distance CNOTs are implemented by distributing the two halves of a maximally entangled state near the target qubits and teleporting the gate.

Appendix B: Error Probability for 3D Cluster States with Fast Entangling Gates

The appendix derives stabilizer-error probabilities for 3D cluster-state creation by tracking independent error sources and finite-range propagation through teleported CNOT links. The bookkeeping separates errors from Bell-pair creation, CNOT links, and final measurements, retaining only linear-order contributions in the local and two-local error strengths.

  • Error propagation: Independent error sources contribute multiplicatively to the stabilizer expectation because the bounded-depth local creation procedure limits propagation to a finite distance.
  • Teleported CNOT links: A teleported CNOT link consists of Bell-state preparation, two CNOT gates, and local Z- and X-basis measurements, realizing a CNOT between the remaining qubits.
  • Error categories: Errors affecting the stabilizer expectation are divided into Type 1 Bell-pair creation, Type 2 CNOT links, and Type 3 final cluster-qubit measurements.
  • CNOT-link errors: Type-2 analysis tracks Z-errors on both face controls and edge targets because edge errors can propagate to neighboring face qubits.
  • Approximation: Only linear-order terms in ϵ and T/τD are retained, with contributions from Bell pairs, two memory-error rounds, CNOT gates, and local measurements.
  • Bell-pair errors: Type-1 errors reduce each face stabilizer factor through the Bell pair created within that face, while neighboring-cell Bell pairs do not contribute directly.
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