Equivalence of continuous- and discrete-variable gate-based quantum computers with finite energy

TL;DR

This work proves that gate-based continuous-variable quantum computers with finite energy cannot surpass discrete-variable quantum computers in computational power, by showing any CV circuit built from a polynomial gate set (Gaussian gates and cubic phase gates) can be efficiently simulated by DV hardware with a controllable, energy-dependent error. The key tool is the stabilizer subsystem decomposition (SSD), which maps CV states and gates to DV qudits, with error bounds that scale with the system energy $E^*$ and qudit dimension $d$. The main theorem provides a concrete, polynomial-time simulation method and an explicit error bound $\epsilon \le 1207 E^{*2} \frac{K n^2}{\sqrt{d}}$, with a corollary enabling qubit-based implementations via binary encoding of qudits. Collectively, these results enable translating CV algorithms to DV or DV-based hardware and imply that, under realistic energy constraints, CV quantum advantage is not exponential; they also offer practical bounds for classical DV simulations of CV circuits and pathways to broader CV-to-DV methodology.

Abstract

We examine the ability of gate-based continuous-variable quantum computers to outperform qubit or discrete-variable quantum computers. Gate-based continuous-variable operations refer to operations constructed using a polynomial sequence of elementary gates from a specific finite set, i.e., those selected from the set of Gaussian operations and cubic phase gates. Our results show that for a fixed energy of the system, there is no superpolynomial computational advantage in using gate-based continuous-variable quantum computers over discrete-variable ones. The proof of this result consists of defining a framework - of independent interest - that maps quantum circuits between the paradigms of continuous- to discrete-variables. This framework allows us to conclude that a realistic gate-based model of continuous-variable quantum computers, consisting of states and operations that have a total energy that is polynomial in the number of modes, can be simulated efficiently using discrete-variable devices. We utilize the stabilizer subsystem decomposition [Shaw et al., PRX Quantum 5, 010331] to map continuous-variable states to discrete-variable counterparts, which allows us to find the error of approximating continuous-variable quantum computers with discrete-variable ones in terms of the energy of the continuous-variable system and the dimension of the corresponding encoding qudits.

Figures (3)

• Figure 1: This figure summarizes the succession of proofs we perform to compare the computational power of RCVQC and DVQC. The boxes in the figure represent the model, and an arrow linking them corresponds to the proof we perform to assess how the connected models can approximate each other. To the left of the arrows, we refer to the Lemma that states the connection.
• Figure 2: In this Figure, we show the circuit diagram consisting of a sequence of operations $\hat{U}$ representing the set $\mathcal{U}_E$. In the top part of the figure, the diagram depicts application of the operation $\hat{U}$ on $n$ number of vacuum input states, and measurements performed in position basis modulo $d\ell$. The lower section of the figure shows the decomposition of $\hat{U}$ in terms of Eq. \ref{['eq: RCVQC gate decomp']}, where we interlace $K$ passive operations and displacements $\hat{V}_j$ with $K$ cubic phase gates, which is then followed by two additional $\hat{V}_j$ operations sandwiching a squeezing operation on each mode. In Fig. \ref{['fig:Universal_circuit_b']}, we show the decomposition of the colored gates in terms of Mach-Zehnder interferometers, rotations, and displacements. The dashed black vertical lines represent time steps. We label the time steps using the notation $t_{i,j}$ where $i$ refers to the round of interlaced Gaussian and cubic phase gates, while $j$ refers to the number of gates applied during the round. Here we denote $M=n(n-1)/2+2n$.
• Figure 3: In this figure, we show the decomposition of an arbitrary colored gate with index $j$ from Fig. \ref{['fig:Universal_circuit_a']}. This decomposition is in terms of a Mach-Zehnder interferometer network according to the design used in Ref. clements2016, where a total of $n(n-1)/2$ Mach-Zehnder interferometers are used with a circuit depth of $n$, which is then followed by rotations and displacements on each mode. The crossings of the wires correspond to the Mach-Zehnder interferometer from Eq. \ref{['eq:Mach-Zehnder']}, whose circuit representation in terms of two rotations and two 50:50 beam splitters is shown at the bottom part of the figure. Each combination of color and wire crossings for the Mach-Zehnder interferometer at different intersections of the grid created by gray dashed lines represents a potentially unique set of angles that parameterize the interferometer.

Theorems & Definitions (42)

• Theorem 1
• Corollary 1
• Definition 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7