Noise Transfer Approach to GKP Quantum Circuits

Abstract

The choice between the Schrödinger and Heisenberg pictures can significantly impact the computational resources needed to solve a problem, even though they are equivalent formulations of quantum mechanics. Here, we present a method for analysing Bosonic quantum circuits based on the Heisenberg picture which allows, under certain conditions, a useful factoring of the evolution into signal and noise contributions, similar way to what can be achieved with classical communication systems. We provide examples which suggest that this approach may be particularly useful in analysing quantum computing systems based on the Gottesman-Kitaev-Preskill (GKP) qubits.

Keywords: cat states Bosonic codes; quantum computing.

Figure 1

Example q quadrature probability distribution for the cat state in Equation (13) with $\alpha =2$.

Figure 2

Example q quadrature probability distribution for the GKP state in Equation (20) with $\mu =1$ and ${\Delta }^2=0.1$.

Figure 3

Example q quadrature probability distribution for the GKP state in Equation (20) with $\mu =1$ and ${\Delta }^2=0.1$ but rotated through a quadrature angle of $\pi /2$. This is equal to the “−” GKP state or, equivalently, the p quadrature probability distribution of the “1” state.

Figure 4

Average position quadrature variance $V_q$ as a function of the parameter $\alpha$ for the cat state defined in Equation (13). Notably, $V_q<1$ for small values of $\alpha$, which can be attributed to clipping effects.

Figure 5

Average position quadrature variance $V_q$ as a function of the squeezing parameter ${\Delta }^2$ for GKP logical states. The computational-basis states are defined in Equation (20), and the dual-basis states are simply rotated versions of the computational-basis states. The dashed line represents ${\Delta }^2$. $V_q$ matches ${\Delta }^2$ for small values of $\Delta$ but deviates in a state-dependent way for larger values. Plotting $V_p$ follows a similar approach, as the p quadrature is simply a rotation, with the computational and dual-basis states switching roles.

Figure 6

Simple teleportation circuit with CZ gates to interact with the modes and feedforward of momentum measurements of mode 1 as imaginary displacements of mode 3 and momentum measurements of mode 2 as real displacements of mode 3. The measurement of mode 1 is represented by the operator ${\widehat{p}}_{1o}$, but if error correction is being implemented, then it is ${\widehat{p}}_{c1o}$, which is fed forward. Similarly, the measurement of mode 2 is represented by the operator ${\widehat{p}}_{2o}$, but if error correction is being implemented, then it is ${\widehat{p}}_{c2o}$, which is fed forward.

Figure 7

The simple teleportation error correction circuit of Figure 3 but with loss errors included for all components. The loss is modelled with beamsplitters, where the transmission of the beamsplitters represents the efficiency of the corresponding components. Additional components (loss and linear amplification of mode 3) are indicated in blue. These components, along with tailored feedforward gains, allow the circuit to still implement error correction. The measurement of mode 1 is represented by the operator ${\widehat{p}}_{1o}$, but if error correction is being implemented, then it is ${\widehat{p}}_{c1o}$ which is fed forward. Similarly, the measurement of mode 2 is represented by the operator ${\widehat{p}}_{2o}$, but if error correction is being implemented, then it is ${\widehat{p}}_{c2o}$ which is fed forward.