Universal Quantum Computation via Superposed Orders of Single-Qubit Gates

TL;DR

The paper shows that universal quantum computation can be achieved by leveraging superposed orders of single-qubit gates via the quantum switch. By constructing a deterministic realization of any two-qubit controlled gate from only single-qubit operations placed in a superposition of causal orders, including the Barenco gate, the authors provide both a general CU framework and explicit recipes for CNOT, CZ, and BAR gates. The approach extends to qudits and is particularly relevant for photonic quantum computing, where the quantum switch is physically realizable. This higher-order computational model opens a new avenue for deterministic quantum computation without relying on pre-shared large entangled resources, potentially impacting fault-tolerant architectures and photonic implementations.

Abstract

Superposed orders of quantum channels have already been proved - both theoretically and experimentally - to enable unparalleled opportunities in the quantum communication domain. As a matter of fact, superposition of orders can be exploited within the quantum computing domain as well, by relaxing the (traditional) assumption underlying quantum computation about applying gates in a well-defined causal order. In this context, we address a fundamental question arising with quantum computing: whether superposed orders of single-qubit gates can enable universal quantum computation. As shown in this paper, the answer to this key question is a definitive "yes". Indeed, we prove that any two-qubit controlled quantum gate can be deterministically realized, including the so-called Barenco gate that alone enables universal quantum computation.

Figures (10)

• Figure 1: Simplified diagram of a linear optic implementation of a $\mathtt{CNOT}$ logic gate, where $Q_0$ and $Q_1$ represent the control qubit and the target qubit and $a_0$, $a_1$ are two ancillary photons.
• Figure 2: KLM $\mathtt{CNOT}$ scheme with simplified NS gates. $Q_0$ and $Q_1$ denote the control qubit and the target qubit, respectively. This scheme assumes the logical qubits to be encoded through spatial modes (path encoding).
• Figure 4: Schematic diagram of some of the experimentally implemented architectures of the photonic quantum switch. (a) An implementation via a Mach-Zehnder geometry, where the target qubit is encoded in polarization of the photon, while the control qubit is mapped into its path degree of freedom using the first beam splitter and coherently recombining the paths $\mathcal{A} \rightarrow \mathcal{B}$ and $\mathcal{B} \rightarrow \mathcal{A}$ at the second beam splitter ProMoqAra-15RubRozFei-17Pro-19Guo2020RubTozMas-22. (b) An implementation via a Sagnac geometry, where the target qubit is encoded in polarization of the photon (as in (a)), whereas a single beam splitter introduces the path degree of freedom as control and completes superposition of causal orders of $\mathcal{A}$ and $\mathcal{B}$StrSchPet-22. (c) An implementation via a geometry, where the target qubit is encoded in the path degree of freedom of the photon, while the role of the control qubit is played by its polarization GosGiaKew-18GosRomWhi-18.
• Figure 5: The $\mathtt{CU}$ (controlled-$\mathtt{U}$) logic gate.
• Figure 6: Representation of two-qubit controlled logic gate via quantum switch, with the switch represented as a H-shape blue box as in Oreshkov2019KriChiSal-20Milz2022LugBarChi-23. The quantum switch combines two-qubit gates $A = A_0 \otimes A_1$ and $B = B_0 \otimes B_1$ in superposed orders, with $A_0,B_0$ denoting the single-qubit unitaries acting on the first qubit $Q_0$ and $A_1,B_1$ denoting the single-qubit unitaries acting on the second qubit $Q_1$.

Theorems & Definitions (18)

• Lemma 1
• proof
• Corollary 1
• proof
• Definition 1
• Definition 2
• Definition 3
• Lemma 2
• proof
• Theorem 1