Universal Compiling and (No-)Free-Lunch Theorems for Continuous-Variable Quantum Learning

Quantum compiling, where a parameterized quantum circuit is trained to learn a target unitary, is an important primitive for quantum computing that can be used as a subroutine to obtain optimal circuits or as a tomographic tool to study the dynamics of an experimental system. While much attention has been paid to quantum compiling on discrete-variable hardware, less has been paid to compiling in the continuous-variable paradigm. Here we motivate several, closely related, short-depth continuous-variable algorithms for quantum compilation. We analyze the trainability of our proposed cost functions and numerically demonstrate our algorithms by learning arbitrary Gaussian operations and Kerr nonlinearities. We further make connections between this framework and quantum learning theory in the continuous-variable setting by deriving no-free-lunch theorems. These generalization bounds demonstrate a linear resource reduction for learning Gaussian unitaries using entangled coherent-Fock states and an exponential resource reduction for learning arbitrary unitaries using two-mode-squeezed states.

Popular Summary

Quantum compiling, where a parameterized quantum circuit is trained to learn a target unitary, is an important quantum algorithm for quantum computing. It can be used to find optimal (i.e., short-depth and noise-resistant) circuits to aid the implementation of larger algorithms, or as a tool to learn the dynamics of an unknown experimental system. Here we develop a framework for the compiling of continuous-variable (CV) operations using CV quantum computers, where information is stored in continuous variables such as the position or momentum of a photonic system.

Our framework is composed of two main components. Firstly, we introduce a series of quantum algorithms for compiling CV operations. Our algorithms are variational quantum algorithms, where a cost function is evaluated on a quantum computer, while a classical optimizer trains a parameterized quantum circuit to minimize this cost. The cost is chosen such that the parameterized quantum circuit that minimizes our costs is guaranteed to match the target unitary we are trying to compile. We demonstrate that our algorithms work by numerically simulating the learning of Gaussian operations, the beam splitter operation, and Kerr nonlinearities. Secondly, we place the field on rigorous foundations by deriving no-free-lunch theorems on the resources required to learn different classes of CV operations.

Our results highlight that entanglement is an important resource for the learning of CV operations. This work paves the way for implementing substantial algorithms on photonic quantum computers. More generally, we hope our quantum compilation algorithms will be used as a novel technique for studying the properties of optical materials.

Figure 1

Learning experimental CV quantum systems. Here we sketch an experimental circuit to learn the unitary $U$ implemented by a novel optical material (shown in orange) by training a parameterized quantum circuit $V^{†}(\theta )$ implemented on an optical quantum computer (shown in gray). For specific details on the proposed circuit, and the cost it calculates, see Fig. 2.

Figure 2

Schematic of cost functions. In this figure we show how the three different cost functions we propose for CV quantum compiling (d)–(f) are related to three different ways of measuring the Hilbert-Schmidt test cost for DV compilation (a)–(c). The probability of measuring the all-zero state on all $2n$ qubits in (a) and (b) is equal to $\mathrm{Tr}[U{V}^{†}]/d^2$ and hence can be used to compute $C_{\mathrm{HST}}$. In (c) $W_{{\psi }_j}$ is the unitary that prepares the state $|{\psi }_j⟩$, i.e., $W_{{\psi }_j}|0⟩=|{\psi }_j⟩$; therefore, the probability of measuring the all-zero state on $n$ qubits is equal to $|⟨{\psi }_j|U{V}^{†}|{\psi }_j⟩{|}^2$. We could theoretically estimate $C_{\mathrm{HST}}$ by running this circuit over a Haar random ensemble of training states, but for large problems, this is exponentially inefficient. In (d) and (e) $S(r)$ is the unitary single-mode squeeze operator and $⟩\bullet ⟨$ is a 50:50 beam splitter that entangles the squeezed registers. The probability of measuring the all-zero state on all $2m$ modes in (d) and (e) is equal to $1-C_{\mathrm{LE}\text{−}\text{TMSS}}$ and $1-C_{\mathrm{R}\text{−}\mathrm{TMSS}}$, respectively. In (f) $D(\alpha )$ is the unitary single-mode displacement operator. The mean probability of obtaining the all-zero state on $m$ modes is equal to the average of the $k$ values $|⟨{\alpha }_j|U{V}^{†}|{\alpha }_j⟩{|}^2$, which is used to estimate $C_{\mathrm{ACS}}$.

Figure 3

Local cost functions. Here we show the circuit diagrams for the local versions of the (a) Loschmidt echo TMSS cost $C_{\mathrm{LE}\text{−}\text{TMSS}}^{(L)}$, (b) ricocheted TMSS cost $C_{\mathrm{R}\text{−}\mathrm{TMSS}}^{(L)}$, and (c) averaged coherent state cost $C_{\mathrm{ACS}}^{(L)}$. Crucially, in contrast to their respective global variants, only a pair of modes (in the case of $C_{\mathrm{LE}\text{−}\text{TMSS}}^{(L)}$ and $C_{\mathrm{R}\text{−}\mathrm{TMSS}}^{(L)}$) and a single mode (in the case of $C_{{\mathrm{ACS}}_E}^{(L)}$) is measured per circuit evaluation.

Figure 4

Cost landscapes. The cost landscape for learning (a) an arbitrary Gaussian operation and (b) a $\chi =3$ Kerr nonlinearity using four layers of the general ansatz defined in Eq. (31). Here $ϵ$ is a noise parameter that determines the deviation of the ansatz parameters, $\mathit{\theta }$, from the optimum parameters, ${\mathit{\theta }}^{\mathrm{opt}}$. Specifically, we set ${\theta }_k={\theta }_k^{\mathrm{opt}}+ϵR$, where $R$ is a random number between $-1$ and 1.

Figure 5

Learning using TMSS cost. The $C_{{\mathrm{LE}\text{−}\text{TMSS}}_r}$ cost (blue) as a function of iteration for learning a Gaussian (top row), Kerr nonlinearity (middle row), and beam splitter (bottom row). To mitigate the problem of vanishing cost gradients for large $r$, we take a perturbative approach starting with $r=0.1$ (left column), and after convergence increasing $r$ to $0.5$ (middle column) and then $2.5$ (right column). To quantify the quality of the optimization, we take the optimal parameters obtained at each iteration of the optimization algorithm and plot both the Hilbert-Schmidt cost $C_{\mathrm{HST}}$ (red dashed) and the errors in the individual optimized parameters (dotted). The parameter errors are given in natural units with $\hslash =1$ and the mass and frequency of the modes equal to 1.

Figure 6

Learning Gaussian operations using coherent states and entangled coherent-Fock states. Cost as a function of iteration for learning a Gaussian operation. In the left and middle plots we use $C_{\mathrm{ACS}}$ with a single training pair ($k=1$) and two training pairs ($k=2$), respectively. In the right plot we optimize $C_{\mathrm{ECFS}}^{(\mathfrak{r}=2,k=1)}(V,U)$ that is defined in Eq. (47) from one training state of the form (44). In all cases the coherent states trained on had a random energy up to a maximum of $1$ and a random phase in the range $0$ to $2\pi$. To quantify the quality of the optimization, we take the optimal parameters obtained at each iteration of the optimization algorithm and plot both the Hilbert-Schmidt cost $C_{\mathrm{HST}}$ (red) and the errors in the individual optimized parameters in arbitrary units (dotted). The dashed and dash-dot red lines show the Hilbert-Schmidt cost $C_{\mathrm{HST}}$ on the first $50$ Fock states and on the first 5 Fock states, respectively.

Figure 7

Learning Kerr nonlinearities using coherent states and entangled coherent-Fock states. Cost as a function of iteration for learning a $\chi =0.1$ (top) and $\chi =0.5$ (bottom) nonlinearity. In the left and middle columns we use $C_{\mathrm{ACS}}$ with a single training pair ($k=1$) and two training pairs ($k=2$), respectively. In the right-hand column we use $C_{\mathrm{ECFS}}$ with a single entangled training pair ($k=1$ and $\mathfrak{r}=2$). In all cases the basic coherent state training states have a random energy up to a maximum of $1$ and a random phase in the range $0$ to $2\pi$. To quantify the quality of the optimization, we take the optimal parameters obtained at each iteration of the optimization algorithm and plot both the Hilbert-Schmidt cost $C_{\mathrm{HST}}$ (red) and the errors in the individual optimized parameters in arbitrary units (dotted). The dashed red line and dash-dot red lines show the truncated Hilbert-Schmidt cost $C_{\mathrm{HST}}$ on the first $50$ Fock states and on the first $5$ Fock states, respectively.