Advancing Quantum Networking: Some Tools and Protocols for Ideal and Noisy Photonic Systems

TL;DR

It is proved, translating a result from representation theory, that there are no non-universal infinite closed 2-designs in U(V) when dim V ≥ 2, and it is observed that linear optical unitaries form a 1-design but not a 2-design.

Abstract

Quantum networking at many scales will be critical to future quantum technologies and experiments on quantum systems. Photonic links enable quantum networking. They will connect co-located quantum processors to enable large-scale quantum computers, provide links between distant quantum computers to support distributed, delegated, and blind quantum computing, and will link distant nodes in space enabling new tests of fundamental physics. Here, we discuss recent work advancing photonic tools and protocols that support quantum networking. We provide analytical results and numerics for the effect of distinguishability errors on key photonic circuits; we considered a variety of error models and developed new metrics for benchmarking the quality of generated photonic states. We review a distillation protocol by one of the authors that mitigates distinguishability errors. We also review recent results by a subset of the authors on the efficient simulation of photonic circuits via approximation by coherent states. We study some interactions between the theory of universal sets, unitary t-designs, and photonics: while many of the results we state in this direction may be known to experts, we aim to bring them to the attention of the broader quantum information science community and to phrase them in ways that are more familiar to this community. We prove, translating a result from representation theory, that there are no non-universal infinite closed $2$-designs in $U(V)$ when $\dim V \geq 2$. As a consequence, we observe that linear optical unitaries form a $1$-design but not a 2-design. Finally, we apply a result of Oszmaniec and Zimborás to prove that augmenting the linear optical unitaries with any nontrivial SNAP gate is sufficient to achieve universality.

Figures (8)

• Figure 1: Example of circuit with symmetric losses that commute to the start of the circuit. Instead of applying 16 independent loss channels, it is equivalent to applying 4 loss channels with a modified parameter $\eta \rightarrow \eta^3$. In this diagram, the two qubit 'gates' are beamsplitters, and the boxes with enclosed $\eta$ represent a loss channel with loss rate $1-\eta$.
• Figure 2: Simulating photons with two internal modes. By 'copying' the circuit, additional internal (noise) modes can be simulated. As each circuit copy is independent, these simulations can be carried out separately.
• Figure 3: 3 photon distillation scheme of Ref. marshall_distillation_2022. As indicated, the output is post-selected upon measurement result $(1,1)$. The angles represent the transmission angle in the beamsplitters (see Ref. marshall_distillation_2022 for a description).
• Figure 4: (a) Generalized $X$ measurement. A beamsplitter with transfer matrix $H$ is applied to two modes (generally part of a larger circuit, not pictured). PNRD is then performed on both modes, and we post-select upon measuring exactly $1$ photon total. The figure shows the example with measurement pattern $(1,0)$; pattern $(0,1)$ is also permissible. Often, future steps in a computation will be altered depending on the pattern obtained here. (b) Type $I$ fusion. Dual-rail states are input in each pair of modes (indicated by curly braces). We then apply the generalized $X$ measurement circuit on the two middle modes. Upon successful post-selection, the two unmeasured modes are treated as a single dual-rail qubit. (c) Type $II$ fusion. We input $2$ dual-rail states, then perform $2$ generalized $X$ measurements. If successful, this circuit is equivalent to measuring the pair of commuting two-qubit dual-rail observables $X\otimes X$ and $Z\otimes Z$, with the eigenvalues determined by the obtained measurement patterns. This is discussed in greater generality below.
• Figure 5: The $n$-GHZ state analyzer: Type II fusion is the special case with $n=2$. We apply a Hadamard beamsplitter for each $p\in\mathcal{P}$, then post-select for patterns $m = (m_0, \dots, m_{2n-1})$ with all $m_{2i+1} + m_{2i+2} = 1$ (including $m_{0} + m_{2n-1} = 1$, due to the convention $m_{2n} = m_0$). This operation may be viewed as $n$ generalized $X$ measurements between the modes of $n$ dual-rail input states, where the $n$-GHZ state analyzer is considered successful if and only if all the generalized $X$ measurements are.

Theorems & Definitions (22)

• Lemma 1
• Proposition 1
• Proposition 2
• Lemma 2
• Theorem 1
• Lemma 3
• Remark 3
• Theorem 4
• Remark 5
• Proposition 3