Timing constraints imposed by classical digital control systems on photonic implementations of measurement-based quantum computing

TL;DR

This work analyzes the digital system needed to implement arbitrary one-qubit rotations and controlled-not gates in discrete-variable photonic MBQC, in the presence of an ideal cluster state generator, with the main aim of understanding the timing constraints imposed by the digital logic on the analog system and quantum hardware.

Abstract

Most of the architectural research on photonic implementations of measurement-based quantum computing (MBQC) has focused on the quantum resources involved in the problem with the implicit assumption that these will provide the main constraints on system scaling. However, the `flying-qubit' architecture of photonic MBQC requires specific timing constraints that need to be met by the classical control system. This classical control includes, for example: the amplification of the signals from single-photon detectors to voltage levels compatible with digital systems; the implementation of a control system which converts measurement outcomes into basis settings for measuring subsequent cluster qubits, in accordance with the quantum algorithm being implemented; and the digital-to-analog converter (DAC) and amplifier systems required to set these measurement bases using a fast phase modulator. In this paper, we analyze the digital system needed to implement arbitrary one-qubit rotations and controlled-NOT (CNOT) gates in discrete-variable photonic MBQC, in the presence of an ideal cluster state generator, with the main aim of understanding the timing constraints imposed by the digital logic on the analog system and quantum hardware. We use static timing analysis of a Xilinx FPGA (7 series) to provide a practical upper bound on the speed at which the adaptive measurement processing can be performed, in turn constraining the photonic clock rate of the system. Our work points to the importance of co-designing the classical control system in tandem with the quantum system in order to meet the challenging specifications of a photonic quantum computer.

Figures (11)

• Figure 1: a) A cluster state is made from a rectangular array of qubits (the white dots), each of which may be entangled with its four nearest neighbours. When a computation is performed, a specific pattern of entanglement is required that matches the shape of the circuit. b) The quantum computation is performed by measuring the cluster qubits in bases derived from the measurement pattern. The shaded blue regions show which cluster qubits are involved in implementing which gates. The identity gate is included to pad the length of the one-qubit gate $U=R_x(\zeta)R_z(\eta)R_x(\xi)$ so it matches the CNOT. c) The quantum circuit that is performed by the measurement pattern in b).
• Figure 2: The state of a single qubit can be represented as a point on the Bloch Sphere. A measurement of a single qubit can be made along any straight line through the Bloch sphere. In MBQC, measurements in the purple and green boxes in Figure \ref{['fig:cluster-state']} are made along lines $L$ in the equator of the Bloch sphere, parametrised by a single angle $\phi$. Computational basis measurements (denoted using grey boxes in Figure \ref{['fig:cluster-state']}) are made along the vertical line through $|0\rangle$ and $|1\rangle$.
• Figure 3: The two measurement patterns we consider in this paper are the CNOT gate and the arbitrary one-qubit gate $U=R_x(\zeta)R_z(\eta)R_x(\xi)$. In each block, the black line connected to the top of the box is the adaptive measurement setting $s$. The line connected to the bottom of each box is the measurement outcome $m$. For the CNOT gate, on the left, there are no adaptive measurement settings, because all the measurement bases are $X$ or $Y$. However, the computation of the byproduct operators (shaded in blue) is more complicated, and involves mixing outcomes from the control $C$ and target $T$ rows. On the other hand, for the arbitrary one-qubit gate, the byproduct operator calculation is simple, but the adaptive measurement settings depend on previous measurement outcomes (shaded purple). The commutation correction for each gate is shaded in orange. For the CNOT gate, it involves mixing the byproduct operators before applying the pattern. For the one-qubit gate, the byproduct operators must be stored because they are used in the adaptive measurement setting calculation. In Section \ref{['sec:simple-model']}, the condition is imposed that columns are measured from left to right, so as to be compatible with photonic MBQC.
• Figure 4: a) A single photon in two waveguides can be used as a qubit. If the photon is in the top waveguide, then the qubit is in the $|0\rangle$ state, whereas if it is in the bottom waveguide, the qubit is in the $|1\rangle$ state. b) Computational basis measurements can be performed by placing a single-photon detector at the end of the waveguides. Basic one-qubit operations can be realised using linear optical elements such as c) beamsplitters and d) modulators. Complex operations can be realised by placing the elements one after the other.
• Figure 5: A variable beamsplitter, which realises an $R_x(\phi)$ rotation, is formed by placing two fixed beamsplitters on either side of a modulator.

Theorems & Definitions (1)

• Theorem 1