Digital quantum simulation of beam splitters and squeezing with IBM quantum computers

We present results on the digital quantum simulations of beam-splitter and squeezing interactions. The bosonic Hamiltonians are mapped to qubits and then digitalized in order to implement them in the IBM quantum devices. We use error mitigation and postselection to achieve high-fidelity digital quantum simulations of single-mode and two-mode interactions, as evinced—where possible—by full tomography of the resulting states. We achieve fidelities above 90% in the case of single-mode squeezing with low squeezing values and ranging from 60% to 90% for large squeezing and in the more complex two-mode interactions.

Figure 1

Quantum circuit run in Santiago for the single-mode squeezing unitary (23) and $\widehat{\varepsilon }=0.05$. $q{13}_0, q{13}_1$, and $q{13}_2$ are respectively qubits 1, 0, and 2 in the text.

Figure 2

Final transpiled circuit run in Santiago for the single-mode squeezing unitary (23) and $\widehat{\varepsilon }=0.05$.

Figure 3

Fidelity $F$ results for the digital quantum simulation of single-mode squeezing with squeezing parameter $\widehat{\varepsilon }$ allowing a maximum of two excitations for: Santiago (May/04/2021), using the approximation in Eq. (29) (blue, dashed, circles) and with full tomography (May/19-20/2021) Eq. (27) (orange, dotted, triangles) and Casablanca (May/05/2021), Eq. (29) (red, solid, squares) and (May/19/2021) Eq. (27) (green, dash-dotted, diamonds).

Figure 4

Final transpiled circuit run in $\text{ibmq}\_\text{santiago}$ for the single-mode squeezing unitary allowing four excitations and $\widehat{\varepsilon }=0.1$. We consider two-qubit interactions only, as in Eq. (31). $q{96}_0, q{96}_1, q{96}_2, q{96}_3$, and $q{96}_4$ are respectively qubits 0, 3, 1, 4, and 2 in the text.

Figure 5

Fidelity $F$ results for the digital quantum simulation of single-mode squeezing with squeezing parameter $\widehat{\varepsilon }$ allowing a maximum of four excitations by using Eq. (33): Casablanca (May/05-06/2021) (blue, dashed, circles) and Santiago (May/20-21/2021) (orange, dotted, triangles).

Figure 6

Fidelity $F$ vs number of Trotter steps for the digital quantum simulation of single-mode squeezing allowing a maximum of four excitations in Casablanca (May/05-06/2021) for $\widehat{\varepsilon }=0.125$ (blue, dashed, circles), $\widehat{\varepsilon }=0.175$ (orange, dotted, triangles), $\widehat{\varepsilon }=0.2$ (green, dash-dotted, diamonds), and $\widehat{\varepsilon }=0.3$ (red, solid, squares).

Figure 7

Fidelity $F$ vs number of Trotter steps for the digital quantum simulation of single-mode squeezing allowing a maximum of four excitations in Santiago (May/20-21/2021) for $\widehat{\varepsilon }=0.125$(blue, dashed, circles), $\widehat{\varepsilon }=0.175$ (orange, dotted, triangles), $\widehat{\varepsilon }=0.2$ (green, dash-dotted, diamonds), and $\widehat{\varepsilon }=0.3$ (red, solid, squares).

Figure 8

Sequence of gates that implement the beam-splitting interaction term $U_{+-}^1$ for $\frac{{\varepsilon }_{+-}}{8}=\frac{\pi }{2}$. The order of the qubits is the same as in the text.

Figure 9

Quantum circuit implementing the equivalent of an $X$-basis measurement in the beam-splitter case with one maximum excitation per mode, where each mode can be treated as a qubit.

Figure 10

Quantum circuit implementing the equivalent of a $Y$-basis measurement in the beam-splitter case with one maximum excitation per mode, where each mode can be treated as a qubit.

Figure 11

Full circuit for digital quantum simulation of a beam splitter for ${\varepsilon }_{+-}=\pi /36$ as launched in Santiago. The order of the qubits is the same as in the text.

Figure 12

Final transpiled version of the circuit in Fig. 14.

Figure 13

Full circuit for digital quantum simulation of a beam splitter for ${\varepsilon }_{+-}=\pi /2$ as launched in Casablanca. $q{1}_0, q{1}_1, q{1}_2$ and $q{1}_3$ are respectively qubits 0, 1, 2 and 3 in the text.

Figure 14

Fidelity $F$ results for the digital quantum simulation of a beam splitter (44) with squeezing parameter ${\varepsilon }_{+-}$ in Santiago (blue, dashed, circles) (May/26/2021) and Casablanca (May/24-25/2021) (orange, dotted, triangles).

Figure 15

Fidelity $F$ results for the digital quantum simulation of a beam splitter vs Trotter steps for Santiago (May/26/2021) ${\varepsilon }_{+-}=\pi /36$ (blue, dashed, circles), ${\varepsilon }_{+-}=\pi /2$ (orange, dotted, triangles) and Casablanca (May/26/2021) ${\varepsilon }_{+-}=\pi /36$ (green, dash-dotted, diamonds), ${\varepsilon }_{+-}=\pi /2$ (red, solid, squares).

Figure 16

Full two-mode squeezing transpiled circuit in Casablanca for $\varepsilon =0.1$. $q{60}_0, q{60}_1, q{60}_2$, and $q{60}_3$ are respectively qubits 0, 1, 2, and 3 in the text.

Figure 17

Fidelity results for the digital quantum simulation of two-mode squeezing vs squeezing parameter for Santiago (May/28/2021) (orange, dotted, triangles) and Casablanca (May/29/2021) (blue, dashed, circles).