Fault-Tolerant Linear Optical Quantum Computing with Small-Amplitude Coherent States

Quantum computing using two coherent states as a qubit basis is a proposed alternative architecture with lower overheads but has been questioned as a practical way of performing quantum computing due to the fragility of diagonal states with large coherent amplitudes. We show that using error correction only small amplitudes ($\alpha >1.2$) are required for fault-tolerant quantum computing. We study fault tolerance under the effects of small amplitudes and loss using a Monte Carlo simulation. The first encoding level resources are orders of magnitude lower than the best single photon scheme.

Figure 1

Schematics for unambiguous CSQC (a) $Z$ basis, (b) Bell state measurements, and (c) CSQC teleportation. Thin lines represent modes whose state is a CSQC qubit with the encoding amplitude shown near each line in square brackets. Prepared CSQC qubit states are shown in the logical basis with the boldface font. The $Z$-basis measurement in (a) as described in 8 is performed by determining which mode photons are present. The Bell state measurement in (b) as described in 5 is performed by determining which mode photons are in and how many photons are present. Both these measurements fail when no photons are present. (c) shows how CSQC teleportation [5, 9] is achieved. A Bell state is generated by splitting a $\beta =\sqrt{2}\alpha$ diagonal state on a beam splitter and performing a Bell state measurement on an unknown qubit and one-half of this entangled state. All detectors are photon counters, all beam splitters are $50:50$, and all unlabeled inputs are arbitrary CSQC qubit states.

Figure 2

Schematics for gate entanglement generation with the same layout as in Fig. 1. (a) shows $Z$-rotation entanglement preparation. A $|+⟩$ state with amplitude $\gamma =\sqrt{{\alpha }^2+{\beta }^2+1/2}$ is split at a three way beam splitter (3BS) generating the state $|{\alpha }^{\prime },\alpha ,\beta ⟩+|-{\alpha }^{\prime },-\alpha ,-\beta ⟩$, where ${\alpha }^{\prime }=1/\sqrt{2}$ and $\beta =\alpha$ for the rotation. The ${\alpha }^{\prime }$ mode is mixed at a beam splitter with reflectivity ${cos}^2(\frac{\theta }{2})$ with a coherent state of equal amplitude. The two output modes are then detected and the output is accepted if one photon is measured in total (probability of success approximately 0.3). (b) shows the Hadamard entanglement preparation. Two copies of the entanglement from (a) are used but with different angles $\theta /2=3\pi /4$ and ${\theta }^{\prime }/2=\pi /4$ and one output mode with coherent state amplitude $\beta =\sqrt{1/2}$. Next, one of the $\beta$ modes is combined at a beam splitter with the $\beta$ mode from the other state. The beam splitter has reflectivity ${cos}^2\pi /4$, and the output modes are detected. The generation succeeds when only one photon is detected in total. If we perform an $X$ correction on one of the modes the desired entanglement is produced with a probability of success of approximately 0.04.

Figure 3

Thresholds for CSQC using the 7-qubit Steane and the 23-qubit Golay code for both memory noise models.