Unitary-transformed projective squeezing: applications for circuit-knitting and state-preparation of non-Gaussian states

TL;DR

The paper introduces unitary-transformed projective squeezing to project CV quantum states onto subspaces obtained from squeezed vacua by a unitary. It develops two practical implementations, LCU and VQED, to realize smeared projectors and demonstrates applications to CV entangled states (EPR, cluster) for circuit knitting and to CPS for universal CV computing. Numerical simulations show improved state purity and reduced sensitivity to photon loss, validating the approach as a pathway to larger, more capable CV quantum devices using Gaussian resources and manageable non-Gaussian elements. The framework supports hybrid hardware and offers a versatile route to scalable CV quantum computation with practical experimental prospects.

Abstract

Continuous-variable (CV) quantum computing is a promising candidate for quantum computation because it can, even with one mode, utilize infinite-dimensional Hilbert spaces and can efficiently handle continuous values. Although photonic platforms have been considered as a leading platform for CV computation, hybrid systems that use both qubits and bosonic modes, e.g., superconducting hardware, have shown significant advances because they can prepare non-Gaussian states by utilizing the nonlinear interaction between the qubits and the bosonic modes. However, the size of hybrid hardware is currently restricted. Moreover, the fidelity of the non-Gaussian state is also restricted. This work extends the projective squeezing method to establish a formalism for projecting quantum states onto the states that are unitary-transformed from the squeezed vacuum at the expense of the sampling cost. Based on this formalism, we propose methods for simulating larger quantum devices and projecting states onto the cubic phase state, a typical non-Gaussian state, with a higher squeezing level and higher nonlinearity. To make implementation practical, we can, by leveraging the interactions in hybrid systems of qubits and bosonic modes, apply the smeared projector by using either the linear-combination-of-unitaries or virtual quantum error detection algorithms. We numerically verify the performance of our methods and show that projection can suppress the effect of photon-loss errors.

Figures (13)

• Figure 1: Wigner functions of single-mode states before and after (unitary-transformed) projective squeezing. $\hat{P}_\mathrm{sq}(\gamma)$ and $\hat{P}_\mathrm{CPS}(\gamma,\eta)$ are smeared projectors onto the state with a higher squeezing level of the squeezed vacuum and CPS, respectively. Parameter $\gamma$ determines the increase in squeezing level as introduced in the main text. Here, we consider a 3-dB squeezed vacuum and the CPS developed from a 3-dB squeezed vacuum as initial states and choose parameter $\gamma$ of the smeared projector to increase the squeezing level by 3 dB. (a) Wigner functions of the squeezed vacuum before and after projective squeezing. After projective squeezing, the width of the squeezed vacuum decreases; the squeezing level of the squeezed vacuum rises. (b) Wigner functions of the CPS before and after projective squeezing. After projective squeezing, the stripes of the Wigner function sharpen.
• Figure 2: The circuits to implement the LCU algorithm. (a) The circuit requiring $N_{\mathrm{anc}}$ ancillary qubits. (b) The circuit requiring only one ancillary qubit.
• Figure 3: The circuits to implement the VQED algorithm. Here, $\ket{+}=(\ket{0}+\ket{1})/\sqrt{2}$ is a plus state and $\hat{X}$ ($\hat{Y}$) is a Pauli-$X$ ($Y$) operator in the qubit system. (a) The circuits to implement the "single-mode" VQED algorithm. By executing this circuit, we can obtain the expectation value corresponding to the state $\mathcal{G}_{N_G} \circ {\mathcal{P}}_{l_{N_G}, l'_{N_G}} \circ...\circ \mathcal{G}_1\circ{\mathcal{P}}_{l_1, l'_1} (\hat{\rho}_{\rm in})$ in Eq. \ref{['Eq:vqed']}. Here, we define $\hat{U}_{l_k"}=\hat{U}_{l_k'}\hat{U}_{l_k}^\dagger$. (b) The quantum circuit for implementing the virtual projection onto the entangled subspace. Here, we denote $\hat{U}_{l"}^{(A,B)}=\hat{U}_{l'}^{(A,B)}\hat{U}_{l}^{(A,B) \dag}$.
• Figure 4: Quantum circuits for virtually entangling a two-mode system. Input $\ket{\mathrm{vac}}$ indicates two vacuum states, which are separable states. By iterating this circuit, we can virtually implement the smeared projector $\hat{P}_{\mathrm{Cluster}}(\gamma, g)$. This smeared projector virtually creates the cluster state, which is the entangled state.
• Figure 5: Circuit to implement two-mode CZ' gate teleportation between inputs $\ket{\psi_1}$ and $\ket{\psi_2}$ with the ancillary cluster state, which lies in the gray area. This ancillary state can be virtually created from a two-mode vacuum state with the VQED method in Fig. \ref{['fig:Knitting_Circuits']}, which leads to circuit knitting. Here, $m_1$ and $m_2$ are the outcomes of each measurement. The mathematical formulation of this circuit is detailed in Appendix \ref{['sec:CircuitKnitting']}.