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Resource costs for fault-tolerant linear optical quantum computing
Ying Li, Peter C. Humphreys, Gabriel J. Mendoza, Simon C. Benjamin
TL;DR
LOQC’s nondeterministic entanglement and photon loss make its full hardware costs and component requirements important to quantify. The paper analyzes a fault-tolerant 3D-cluster-state protocol and estimates very large component counts and stringent per-component performance requirements. It identifies optical switches as the largest influence on threshold loss and error rates.
Problem
LOQC lacks deterministic entangling operations, and its many optical components compound photon loss and other errors; prior work had not jointly established per-component requirements and overall resource costs.
Source: §I · p. 1Method
The authors analyze a full linear-optical protocol based on 3D cluster states, tracking component errors, losses, and resource requirements from entanglement primitives through fault-tolerant computation.
Source: §I · p. 1, §II · p. 3Results
The protocol requires on the order of 10^11 detectors, with per-component loss and error thresholds of ∼10^-3 and ∼10^-5, respectively, and component counts several orders greater than systems with deterministic gates.
Source: §VI · p. 7Takeaways & Limitations
For this scheme, optical-switch performance has by far the greatest impact on loss and error thresholds, while other components contribute more equally.
Source: §VI · p. 8
Abstract
from arXiv · showhide
Linear optical quantum computing (LOQC) seems attractively simple: information is borne entirely by light and processed by components such as beam splitters, phase shifters and detectors. However this very simplicity leads to limitations, such as the lack of deterministic entangling operations, which are compensated for by using substantial hardware overheads. Here we quantify the resource costs for full scale LOQC by proposing a specific protocol based on the surface code. With the caveat that our protocol can be further optimised, we report that the required number of physical components is at least five orders of magnitude greater than in comparable matter-based systems. Moreover the resource requirements grow higher if the per-component photon loss rate is worse than one in a thousand, or the per-component noise rate is worse than $10^{-5}$. We identify the performance of switches in the network as the single most influential factor influencing resource scaling.
I. INTRODUCTION
LOQC avoids matter-based information storage but must overcome nondeterministic entanglement and photon loss with complex optical circuits. The paper estimates the full protocol’s resource and component-performance requirements.
I. INTRODUCTION
- LOQC’s lack of deterministic entangling operations and susceptibility to photon loss motivate circuitry that can require more complexity than matter-based approaches.Optical fields offer low decoherence and relatively easy photonic encoding, but switching and delay elements add further errors and losses.Source: §I · p. 1
- The proposed protocol covers steps from entanglement primitives to a scalable fault-tolerant computer, accounting for errors and losses at each stage.It uses a purely linear optical network, assumes on-demand single-photon sources, and is presented as an upper bound because further optimizations may reduce requirements.Source: §I · pp. 1–2
- 10^11–10^12 detectors are required for a 1,000-logical-qubit computer, with per-component loss below ∼10^-3 and error rates below ∼10^-5.The estimate corresponds to upwards of 10^5–10^6 detectors per physically encoded cluster-state qubit.Source: §I · p. 2
II. PROTOCOL
The protocol uses a 3D cluster state for surface-code computation and prepares it from redundant photonic building blocks. Probabilistic links and measurement-based handling of failures allow the cluster to be assembled layer by layer.
II. PROTOCOL
- The 3D cluster-state approach supports surface-code computation, with stated loss and phase-error thresholds that make it suited to LOQC.Without loss, the cluster tolerates phase errors up to 3%; without computational errors, it tolerates up to 24.9% qubit loss.Source: §II · p. 2
- A suitable set of building blocks permits deterministic generation of a fault-tolerant cluster state of arbitrary size, one 2D layer at a time, with factories reused across layers.Each lattice qubit has four neighbours, and one complete building block is prepared for each eventual cluster qubit.Source: §II · p. 3
- Each building block has a photonic core and tree-encoded bridge units; Bell measurements between bridge roots can entangle cores, while tree measurements identify phase corrections after failures with high probability.The bridge structure permits repeated link attempts while keeping errors below the fault-tolerant threshold, provided enough bridge units are available.Source: §II · p. 3
- Four sets of bridge units are allocated to a core’s four neighbours, and a connection fails if no probabilistic entangling operation succeeds or a removal measurement fails.Failed connections are treated as missing qubits, which the 3D-cluster measurement-based computation can tolerate.Source: §II · p. 3
- Core qubits can be measured before cluster preparation because their feedback gates are identity or σz, reducing photon-loss exposure; failed measurements cause the building block to be abandoned.The prepared core state is equivalent to the 3D cluster state up to feedback single-qubit gates.Source: §II · p. 3
III. GENERATION OF BUILDING BLOCKS
Building blocks are assembled from single photons through three-qubit GHZ primitives and probabilistic Bell-measurement fusion. Switch networks and delay lines route successful outputs between stages.
III. GENERATION OF BUILDING BLOCKS
- Three-qubit GHZ states are sequentially combined by Bell measurements to build larger entangled states, a process described as efficient below a photon-loss rate of 1/3.At each stage, the resulting entangled state can increase in size by up to a factor of two.Source: §III · p. 5
- GHZ factories, graph-state joining elements, switching networks, and delay lines are the primary circuit components for constructing building blocks.The operations use linear optical elements such as sources, beam splitters, switches, delay lines, and photon detectors.Source: §III · p. 5
- The protocol uses a six-single-photon GHZ factory with lossless success probability 1/32 and Bell measurements as probabilistic entangling operations.Bell measurements succeed with 50% probability without ancillary resources or 75% with four ancillary photons; Type-I fusion is avoided because it may convert photon loss into computational errors.Source: §III · p. 5
- Successful outputs from probabilistic stages are selected through reconfigurable switchyards, with delay lines allowing time for switchyard reconfiguration.Selecting N successful GHZ copies from M attempts requires six M-input-to-N-output switchyards.Source: §III · p. 5
- A 2-to-2-switch network requires each photon to pass through approximately log2(MN) switches; single-output switchyards reduce loss, while more output modes improve resource use.The protocol evaluates different switch-network configurations to optimize the fault-tolerance threshold.Source: §III · p. 5
IV. PHOTON LOSS AND COMPUTATIONAL ERRORS
The model accounts for photon loss across optical components and computational errors from beam splitters, delay lines, and switches, while excluding higher-order errors from threshold calculations under a low-rate assumption. Figure 4 compares loss thresholds without computational errors across component and switchyard configurations.
IV. PHOTON LOSS AND COMPUTATIONAL ERRORS
- Loss is modeled at single-photon sources, beam splitters, delay lines, switches, and detectors; computational errors are modeled at beam splitters, delay lines, and switches.Imperfect photon-source mode overlap is incorporated into the beam-splitter error rate.Source: §IV · p. 5
- With no computational errors, Fig. 4 plots component-loss thresholds against detectors per data qubit; assuming perfect switches yields the most dramatic improvement.The figure compares switchyard designs and cases where one component type is lossless.Source: Figure 4 · p. 6
- Two-photon emissions, switching errors, and detector dark counts can be corrected, but are omitted from threshold studies because they are assumed much rarer than first-order errors.A two-photon error followed by photon loss can produce a wrong measurement outcome.Source: §IV · pp. 5–6
V. THRESHOLDS
The thresholds depend on building-block resources, component choices, and computational errors. The study finds higher loss tolerance with multi-input/output switches, identifies switching networks as the strongest influence, and reports a component-error threshold on the order of 10^-5.
V. THRESHOLDS
- Larger building blocks improve loss tolerance; detector count is the resource metric because other component totals scale roughly proportionately.Each plotted building block represents one surface-code data qubit, and the study excludes blocks requiring more than two billion detectors.Source: §V · pp. 6–7
- Each Fig. 4 curve is an envelope of tested protocols, representing the best-performing protocols among those simulated.The individual simulation dots are shown only for the upper-left red curve.Source: §V · p. 6
- Threshold loss is ∼0.1%–∼0.2% with cascaded 2-to-2 switches, while multi-input/output switchyards raise the threshold ∼3–5 times.The higher thresholds approach the 1% limiting loss rate discussed for computing with essentially deterministic gates, at additional resource overhead.Source: §V · p. 7
- Switching networks most strongly affect loss tolerance; approaches that extensively recycle intermediate cluster states may face greater switching complexity.The paper relates this potential complexity increase to recycling discussed in Ref.Source: §V · p. 7
- 75%-success Bell measurements perform significantly better than 50%-success measurements, while single-photon-ancilla measurements are slightly better than Bell-state-ancilla measurements.The 75% protocol requires photon-number resolution through 3 rather than 2; more complex ancillary states can offset gains through preparation costs.Source: §V · p. 7
- With computational errors, the threshold error rate per component is on the order of 10^-5, and the threshold loss rate is reduced.Fig. 5 uses cascaded 2-to-2 switches and 75%-success single-photon-ancilla Bell measurements, with equal component loss and error rates.Source: §V · p. 7, Figure 5 · p. 7
VI. DISCUSSION
The proposed 3D-cluster-state LOQC protocol requires component performance beyond the current photonics state of the art and far more components than systems with deterministic gates. Optical switches have by far the greatest impact on its loss and error thresholds.
VI. DISCUSSION
- ∼10^-3 loss and ∼10^-5 error thresholds per component accompany an estimated 10^11 detectors and similar numbers of other components, several orders more than systems with deterministic gates.These thresholds are beyond the current state of the art in photonics.Source: §VI · p. 7
- The authors frame the stringent thresholds as a challenge intended to stimulate further discussion and innovation in LOQC.They also invite theoretical work exploring alternative schemes to improve the thresholds.Source: §VI · pp. 7–8
- Optical-switch performance has by far the greatest impact on the scheme’s loss and error thresholds, while other components contribute more equally.The authors hope this finding will help guide experimental priorities.Source: §VI · p. 8
Appendix A: Loss and error tolerant building blocks
The protocol prepares a tree-encoded building block for each qubit in a 3D cluster state. Redundant branches support probabilistic entanglement and removal operations that tolerate photon loss.
Appendix A: Loss and error tolerant building blocks
- One intermediate building-block state is prepared for each qubit in the 3D cluster state.The cluster state is based on a 3D lattice.Source: Appendix A · p. 9
- Each building block is a star graph with a core photonic qubit and bridge units encoded as tree graph states whose root qubits connect to the core.Each bridge tree contains redundant branches to tolerate photon loss and failed entanglement operations.Source: Appendix A · p. 9
- A probabilistic PEO links two core qubits; success or failure determines which branch qubits are measured in the σz or σx basis.Successful PEOs remove redundant branches, while failed PEOs remove entire bridge units; tree removal can succeed with arbitrarily high probability when photon loss is below 50%.Source: Appendix A · p. 9
Appendix B: Constructing building blocks
Building blocks are assembled from GHZ states through rake and rake-tree graph states, then converted to tree states. The rake structure permits tree growth by single-qubit measurement, reducing construction stages for high-level trees.
Appendix B: Constructing building blocks
- The construction begins with rake states prepared from three-qubit GHZ states, combines them with further GHZ states into rake-tree states, and removes the rake to obtain a building block.The passage illustrates the process with branching numbers (8, 2, 2).Source: Appendix B · p. 9
- 2(r −1) GHZ states and ceil[log2(r−1)]+1 steps prepare an r-branch rake, assuming all Bell measurements succeed.Each step can nearly double the branch count by combining rakes with a PP operation.Source: Appendix B · p. 9
- Rake-tree states grow by increasing tree branching with PP operations or increasing tree level by measuring a rake branch in the σz basis.A rake is a rake-tree state with a one-level tree and branching number zero; a GHZ state is a rake-tree state with one rake branch and a one-level tree of branching number one.Source: Appendix B · p. 9
- A rake permits tree-level growth through a single-qubit measurement, and the largest rake state takes ceil[log2(R −1)] + 1 steps for a final tree of level R.Unlike Ref., this protocol does not require probabilistic Bell measurements to increase tree level; fewer construction stages can reduce delay-line and switchyard noise and resource costs.Source: Appendix B · p. 9
- Bell measurements are used instead of Type-I fusion because photon loss in Type-I fusion may become a computational error.With loss, a one-photon detection can leave the input qubits in VH rather than the expected HH/VV subspace.Source: Appendix B · p. 9
Appendix C: The model of computational errors
The computational-error model describes operations on GHZ states and represents errors using Pauli errors, including correlations introduced by Bell measurements and switchyards. It also characterizes the tree-state branching notation used in the construction process.
Appendix C: The model of computational errors
- Operations on three-qubit GHZ states include Hadamard gates, Bell measurements, and single-qubit measurements in the σx basis.Source: Appendix C · p. 9
- Tree-state branching numbers (b0, b1, b2, …) count bridge units at b0 and branches at each subsequent generation.In the protocol, b0 is always a multiple of four.Source: Figure 6 · p. 10
- Computational errors are represented as Pauli errors on individual or groups of qubits; Bell measurements can introduce correlations between separately prepared GHZ states.The passage identifies X, Y, and Z Pauli errors as the relevant error types.Source: Appendix C · p. 10
- Switchyard-induced correlations are managed by assigning states from one switchyard to cluster-state qubits separated by distances much larger than a logical qubit.The stated aim is to prevent those correlations from forming an error that risks a logical qubit.Source: Appendix C · p. 10
Appendix D: Numerical simulations
Appendix D describes how loss thresholds are estimated across building-block and switch-network designs, and how computational errors are evaluated. The threshold estimate uses the stated loss-only and phase-error-only limits for 3D cluster states.
Appendix D: Numerical simulations
- The estimated 3D-cluster threshold combines loss and phase error using loss-only and phase-error-only thresholds of 24.9% and 2.93%, respectively.The simulations evaluate the loss rate pL and phase-error rate pP of a cluster-state qubit for given per-component rates.Source: Appendix D · p. 11
- The loss-only study spans 8,400 tree structures and 627 switch-network configurations, with each fancy switchyard’s output/input ratio set to its operation’s success rate including photon loss.Tree branching numbers are at most 20; the 627 configurations use 2-input-2-output switches.Source: Appendix D · p. 11
- For the plotted curves, thresholds are evaluated across 8,400 × 627 protocols for most cases, while Fig. 4(b) varies only the 8,400 building-block structures.Each curve is the envelope of the evaluated thresholds.Source: Appendix D · p. 11
- With computational errors, about 500 near-envelope protocols requiring no more than 2 × 10^9 detectors per case are evaluated using 100,000 Monte Carlo samples per loss- and error-rate setting.The selected protocols are chosen for strong loss tolerance; their cluster-state phase-error rates are estimated by Monte Carlo simulation.Source: Appendix D · p. 11
Appendix E: Bell measurements with entangled ancillary states
Appendix E compares Bell measurements assisted by entangled states and tree-structure choices for their loss thresholds. The authors report that further boosting Bell-measurement success with more entangled ancillary photons is not helpful, and that 4-level trees tolerate less loss in the studied regime.
Appendix E: Bell measurements with entangled ancillary states
- With fancy switches and no computational errors, 4-level trees tolerate less photon loss per component than 2- and 3-level trees in the studied regime.Fig. 8 presents loss-rate thresholds for designs with equal component loss rates.Source: Figure 8 · p. 12
- Further boosting Bell-measurement success with more entangled ancillary photons is not helpful, despite the comparison between 75% and 87.5% success protocols.The 87.5% protocol uses a 4-qubit GHZ state prepared from two 3-qubit GHZ states.Source: Appendix E · p. 11, Figure 9 · p. 12
- The Fig. 9 thresholds compare equal-component-loss designs using Bell measurements with 75% success assisted by a Bell state and 87.5% assisted by a 4-qubit GHZ state.The figure describes thresholds without computational errors.Source: Figure 9 · p. 12
- Fig. 10 sets equal component loss rates and computational error rates of 3 × 10^-6, while Fig. 11 plots loss thresholds across error rates and detector counts from 10^5 to 10^9 per data qubit.In Fig. 11, detector count increases from bottom to top; fluctuations reflect Monte Carlo uncertainty.Source: Figure 10 · p. 12, Figure 11 · p. 13