Continuous-Variable Instantaneous Quantum Computing is Hard to Sample

Instantaneous quantum computing is a subuniversal quantum complexity class, whose circuits have proven to be hard to simulate classically in the discrete-variable realm. We extend this proof to the continuous-variable (CV) domain by using squeezed states and homodyne detection, and by exploring the properties of postselected circuits. In order to treat postselection in CVs, we consider finitely resolved homodyne detectors, corresponding to a realistic scheme based on discrete probability distributions of the measurement outcomes. The unavoidable errors stemming from the use of finitely squeezed states are suppressed through a qubit-into-oscillator Gottesman-Kitaev-Preskill encoding of quantum information, which was previously shown to enable fault-tolerant CV quantum computation. Finally, we show that, in order to render postselected computational classes in CVs meaningful, a logarithmic scaling of the squeezing parameter with the circuit size is necessary, translating into a polynomial scaling of the input energy.

Figure 1

Left: IQP circuit on $n$ qubits. $|+⟩$ is the $\widehat{X}$ eigenstate associated with eigenvalue $+1$. Measurements are performed in the $\{|\pm ⟩\}$ basis. We denote $D_Z(n)=\prod _{z\in {\mathbb{Z}}_2^n}\exp (i\theta (z,n)\underset{j=1}{\overset{n}{\otimes }}{Z}^{z_j})$. Right: IQP circuit in CVs. $|\sigma {⟩}_p$ are finitely squeezed states with variance $\sigma$ in the $\widehat{p}$ representation and $|{\widetilde{+}}_L⟩$ are finitely squeezed GKP states. The gate ${\widehat{D}}_q$ is a uniform combination of elementary gates from the set mentioned in the text. The finitely resolved homodyne measurement ${\widehat{p}}^{\eta }$ has resolution $2\eta$.

Figure 2

Left: Hadamard gadget in a postselected IQP circuit, where $h$ takes value 0 if $+1$ is measured, while $h=1$ if the result is $-1$. Right: Ideal Fourier gadget in CVs, exact translation of the Hadamard gadget. $|0{⟩}_p$ represents an infinitely $\widehat{p}$-squeezed state with $\sigma =0$, thus, satisfying $\widehat{p}|0{⟩}_p=0$.

Figure 3

Procedure to correct for errors in the $\widehat{q}$ quadrature. $|\psi ⟩$ is the data qubit and $|{\tilde{0}}_L⟩$ is a realistic, i.e., noisy, GKP state. After measurement on the second mode, the result $p_k$ is used to shift the first mode back.