Universality of a standard two-qubit gate by catalytic embedding

TL;DR

The paper shows that the two-qubit gate $CV$ is computationally universal when augmented with catalytic embeddings, enabling simulation of standard universal gate sets with constant overhead. It provides a concrete three-step catalytic-embedding protocol to simulate $V$, $S$, and $T$, achieving exact simulation for certain unitary classes and approximate synthesis for others using resource states $|1\rangle$, $| -\rangle$, and $|T\rangle$. By combining this simulation with existing synthesis methods, it yields exact synthesis for unitaries with entries in $\mathbb{Z}[\tfrac{1}{2},i]$ and approximate Clifford$+T$ synthesis for units in $\mathbb{Z}[\tfrac{1}{2},\omega]$, with a feasible resource overhead and limited ancilla. The results imply that universal quantum computation can emerge from a comparatively simple, hardware-friendly gate like $CV$, paired with catalytic-state resources and standard classical-like synthesis techniques, potentially enabling fault-tolerant implementations on near-term devices.

Abstract

We study the resources required to achieve universal quantum computing via the gate sets that provide the fundamental instructions from which quantum algorithms are built. While single-gate universal sets are known, they rely on precisely tuned irrational rotations, making them difficult to realize on near-term devices. We find that the controlled-$V$ gate, an elementary two-qubit interaction directly implementable on leading hardware, is universal and capable of simulating standard universal gate sets with minimal overhead. Specifically, we use catalytic embeddings to develop a constant-overhead algorithm that simulates standard universal gate sets, including Clifford$+T$ and Clifford$+$Toffoli. We combine this simulation algorithm with existing synthesis results to yield exact and approximate synthesis algorithms for unitaries with and without number-theoretic restrictions. The results highlight how full quantum computational power, complete with algorithms for synthesis and simulation, can emerge from unexpectedly simple ingredients.

Figures (3)

• Figure 1: We assume that both nearest-neighbour configurations of the $CV$ gate shown in (a) and (b) are available, and use these to derive (c) $CX$, (d) $CV^\dagger$, (e) $\mathit{SWAP}$, and (f) the Toffoli gate by the Sleator-Weinfurter construction PhysRevA.52.3457.
• Figure 2: The encoding of the gates $V$, $S$, and $T$ is defined by the function $\mathcal{E}(-)$ is shown in (a), relying on three named auxiliary qubits $\alpha$, $\beta$, and $\gamma$. The encoding of $V$ and $S$ in turn allows one to derive (b) the Hadamard gate (up to a global phase) and (c) the controlled $S$ gate, and (d) the controlled Hadamard gate using standard circuit identities. The encoding of $T$ relies on the encoding of $V$ and $S$ (and subsequent derivations of $H$ and $CS$).
• Figure 3: Circuits related to the encoding: (a) The $CV$ circuit encoding a $CS$ gate by decomposing it into $V$, $S$, and $CV$ gates using the equations in Fig. \ref{['fig:encoding']} and encoding the $V$ and $S$ gates. The auxiliary qubit $\gamma$ is unused and has been omitted. (b) The circuit producing a $\ket-$ state when the measurement outcome is $1$ (corresponding to the eigenstate $\ket0$), which occurs with probability $\tfrac{1}{2}$.