Resource Costs for Fault-Tolerant Linear Optical Quantum Computing

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 optimized, we report that the required number of physical components is at least 5 orders of magnitude greater than in comparable matter-based systems. Moreover, the resource requirements grow further if the per-component photon-loss rate is worse than $10^{-3}$ 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.

Popular Summary

In the decades since Richard Feynman proposed the possibility of a quantum computer, physicists have examined many different ways to build this “ultimate machine.” One elegant possibility is to use particles of light—photons—as the sole means of representing and processing information stored as electromagnetic degrees of freedom, such as polarization. Calculations can then be performed using basic, well-understood components such as lenses, mirrors, beam splitters, and detectors. This setup represents linear optical quantum computing. But can such a seemingly simple solution be truly practical? Here, we conduct, for the first time, a detailed theoretical study aimed at shedding light on this question.

We focus on pure linear optical quantum computing, and we assume that single photons can be generated reliably. These kinds of systems pay a unique price for their simplicity. When qubits are represented as photons, operations between qubits cannot be performed “deterministically”; there is always a strong possibility that the operation will fail. This possibility of failure can be overcome by attempting the same task multiple times and routing forward only successful outcomes, which must be achieved while controlling photon losses (from beam splitters, detectors, and switches, for example) and noise. We find that photon-based devices must have approximately 100,000 more physical components than matter-based systems using trapped atoms or superconducting circuits.

Our results do not rule out all-optical quantum computers, but they do reveal how demanding the requirements are for achieving these “ultimate machines.” We anticipate that our work will invigorate the linear optical quantum computing community to address the fidelity targets or derive better protocols.

Figure 1

Protocol for linear optical quantum computing using 3D cluster states. This figure uses the graph-state notation: Each diagram represents the multiqubit state that would result if one could prepare one qubit in state $|+⟩$ for each node (filled circle) and then perform a controlled-phase gate for each edge (i.e., connecting line). However, since LOQC does not permit deterministic entanglement, our states must actually be created by a more lengthly process. (a) The full 3D cluster state is constructed in a near-deterministic step once we have created sufficiently complex building-block states. Each building-block state contains a photonic core qubit (solid circle) and several bridge units (empty circles), which are themselves tree graph states of photonic qubits. The core qubits of the building-block states will form the qubits in the 3D cluster state. Entanglement between these cores (i.e., edges in the eventual cluster state) is established by attempting PEOs between bridge units. (b) PEOs on two bridge units are used to make entanglement links between core qubits. The PEO is composed of a Hadamard gate on one of the bridge’s root qubits followed by a Bell measurement (BM) between the two root qubits. Other photonic qubits in the bridge qubit are then measured in bases according to the outcome of the BM. If the BM succeeds, an entanglement link is generated. Regardless of whether it succeeds or fails, the remains of the two bridge units must be removed from their building blocks.

Figure 2

Building-block states are constructed in a series of stages from initial three-qubit GHZ-state entanglement resources. Further details are given in Appendix pp2.

Figure 3

(a) Circuit for realizing linear optical quantum computing using a three-dimensional cluster state. The circuit includes stages for (i) generating initial three-qubit GHZ states, (ii) synthesizing these states into building-block states, and finally (iii) constructing the 3D cluster state. Qubits on the cluster state are physically measured right after they are generated by a GHZ-state factory. (b) GHZ-state factories probabilistically generate three-qubit GHZ states from six single photons. Successful generation of the GHZ state is heralded by specific three-photon detection events at the detectors. To select at most $N$ successful copies of the GHZ state from $M$ attempts, we need six $M$-to-$N$ switchyards. Delay lines are necessary before photons enter switchyards to allow for feed forward. (c) Bell-measurement circuits synthesize building-block states. The two different circuits depicted succeed with probabilities 50% and 75%, respectively. For the 75%-success circuit, four ancillary single photons are used. At each synthesis stage, many copies of input states are prepared, and measurements are performed on these states in parallel. Successful output states are selected with switch networks. (d) A $M$-to-$N$ switchyard can be realized with one-to-$N$ switchyards and $M$-to-one switchyards. For each one-to-$N$ switchyard, every output mode is connected to an input mode of a different $M$-to-one switchyard. A $M$-to-one switchyard is composed of approximately $M$ two-to-two switches. A one-to-$N$ switchyard is similar.

Figure 4

Thresholds on the loss rate per component $p$ as a function of the number of detectors per data qubit in the case of no computational errors; $p_S$ denotes the success rate of Bell measurements. We again note that each logical qubit will be encoded in a surface code consisting of $\ge 1000$ data qubits. (a) Model in which each switchyard is composed of cascaded two-to-two switches. All components have the same loss rate, i.e., $p_e=p_b=p_d=p_s=p_m=p$. (b) Each switchyard is a single switch with multiple inputs and outputs. All components have the same loss rate $p$. (c)–(f) In each plot, one form of the component is assumed to be perfect, while all other components have the same loss rate $p$. As can be seen, the most dramatic improvement is seen when the switches are assumed to be perfect. Transparent curves in subfigure (f) correspond to the case that the loss rate of switches is 10% of other components.

Figure 5

Thresholds on the loss rate per component $p$ for given the error rate per component $\epsilon$ and the number of detectors per data qubit. Switchyards are composed of cascaded two-to-two switches, and single-photon ancilla-assisted Bell measurements with 75%-success probability are used. All components have the same loss rate, i.e., $p_e=p_b=p_d=p_s=p_m=p$, and all error rates are equal, i.e., ${\epsilon }_b={\epsilon }_d={\epsilon }_s=\epsilon$. See Fig. 11 for the full data with more details.

Figure 6

Scheme for generating building-block states with three-qubit GHZ states. The building-block state is a tree graph state of photonic qubits. Such a tree can be characterized by branching numbers $(b_0,b_1,b_2,\dots )$, where $b_0$ is the number of bridge units and $b_i$ is the number of $i$th-generation branches of a bridge unit. In our protocol, $b_0$ is always a multiple of 4. Graph states occurring in the generation process include three-qubit GHZ states, rake states, tree states, and rake-tree states. On the top of the figure, the example rake state has three branches, the rake-tree state is composed of a rake with three branches and a two-level tree with branching numbers (2,2), and the building-block state is a three-level tree state with branching numbers (8,2,2).

Figure 7

Fusion operations on graph states using Bell measurements in the basis $|{\mathrm{BS}}_{\mu ,\nu }⟩=Z_{a^{\prime }}^{\mu }{X}_{b^{\prime }}^{\nu }(|00{⟩}_{a^{\prime },b^{\prime }}+|11{⟩}_{a^{\prime },b^{\prime }})/\sqrt{2}$, where $\mu ,\nu =0,1$. (a) A CP operation includes a Hadamard gate on one of two measured qubits and then the Bell measurement. Depending on the measurement outcome, an operation $Z_a^{\nu }{Z}_b^{\mu }$ needs to be performed. (b) A PP operation includes the Bell measurement and a Hadamard gate on the nearest neighboring qubit of one of two measured qubits. Depending on the measurement outcome, an operation $Z_a^{\nu }{Z}_b^{\mu }{\prod }_{i\in N^{\prime }(b)}{Z}_i^{\mu }$ need to be performed. Here, $N^{\prime }(c)$ denotes neighboring qubits of the qubit $c$ except the qubit $c^{\prime }$.

Figure 8

Thresholds of the loss rate per component without computational errors for a computer built with fancy switches. Panels (a)–(c) correspond to different values of $p_s$ as labelled in the figure. All components have the same loss rate. In addition to two-level and three-level trees, we have also considered four-level trees. One can find that four-level trees tolerate less photon loss per component within the regime in which we are interested.

Figure 9

Thresholds of the loss rate per component without computational errors using Bell measurements with the success probability 75% (assisted by a Bell state) and the success probability 87.5% (assisted by a four-qubit GHZ state). Panel (a) corresponds to the use of simple switches while panel (b) corresponds to fancy switches, i.e. switches with arbitrarily large numbers of inputs and outputs. All components have the same loss rate.

Figure 10

Thresholds of the loss rate per component $p$ as a function of the number of detectors per data qubit in the case with computational errors. Panel (a) corresponds to the use of simple switches while panel (b) corresponds to fancy switches. All components have the same loss rate, i.e., $p_e=p_b=p_d=p_s=p_m=p$, and the rate of computational errors per component is ${\epsilon }_b={\epsilon }_d={\epsilon }_s=3\times 10^{-6}$.

Figure 11

(a) The full data of Fig. 5. (b) Thresholds on the loss rate per component $p$ given the error rate per component $\epsilon$ with ${10}^5,{10}^6,\dots ,{10}^9$ detectors per data qubit from bottom to top. Fluctuations are due to the uncertainty of the Monte Carlo method.