Squeezing and quantum approximate optimization

Variational quantum algorithms offer fascinating prospects for the solution of combinatorial optimization problems using digital quantum computers. However, the achievable performance in such algorithms and the role of quantum correlations therein remain unclear. Here, we shed light on this open issue by establishing a tight connection to the seemingly unrelated field of quantum metrology: Metrological applications employ quantum states of spin ensembles with a reduced variance to achieve an increased sensitivity, and we cast the generation of such squeezed states in the form of finding optimal solutions to a combinatorial MaxCut problem with an increased precision. By solving this optimization problem with a quantum approximate optimization algorithm (QAOA), we show numerically as well as on an IBM Quantum chip how highly squeezed states are generated in a systematic procedure that can be adapted to a wide variety of quantum machines. Moreover, squeezing tailored for the QAOA of the MaxCut permits us to propose a figure of merit for future hardware benchmarks. Exploiting the connection, we show how the performance can be improved by warm-starting the optimization algorithm with the squeezed state.

Figure 1

Metrologically useful squeezing generated by a depth-three QAOA for the MaxCut problem in a complete graph of 12 nodes. (a) Fully connected unweighted graph with the nodes and edges colored according to one of the 924 possible maximum cut configurations. (b) Wigner quasiprobability distribution of the symmetric Dicke state $D_6^{12}$, an idealized example of a squeezed state and the target for our QAOA. (c) Circuit representation of QAOA with alternating application of cost function and mixer Hamiltonian. The Bloch spheres and histograms from (d) to (j) show the state after the corresponding gate in the optimized QAOA circuit with $({\gamma }_i,{\beta }_i)$ given by $(0.199,0.127), (0.306,0.087)$, and $(4.592,1.518)$ for $i=1$, 2, and 3, respectively. Negativity in the Wigner distribution indicates that the states are non-Gaussian [27]. The squeezing (in black), energy expectation $\langle {\widehat{H}}_C\rangle$ (in blue), and overlap probability density $|\langle D_6^{12}|\psi \rangle {|}^2$ with the target Dicke state (in orange) are shown inside each histogram.

Figure 2

Example of a six-qubit quantum volume circuit as presented in Ref. [21], which shows the layers of $\mathrm{SU}\left(4\right)$ and swap gates.

Figure 3

Benchmarking QAOA with squeezing. (a) Probability distribution ($p_m$, red solid line) and normalized cut size ($c_m/c_{\text{max}}$, blue dashed line) simultaneously plotted against $m=\langle {\widehat{L}}_{\text{z}}\rangle +\frac{n}{2}$ for $n=30$. States with normalized cut size more than $\alpha$ lie in $m\in [\lceil m_{-}\left(\alpha \right)\rceil ,\lfloor m_{+}\left(\alpha \right)\rfloor ]$. These yield the shaded area under the probability $p_m$, which is the figure of merit $P_{\alpha }$ defined in Eq. (11). (b) $P_{\alpha }$ showing how the probability of sampling high-value cuts changes with the squeezing $\mathcal{S}$ and the number of qubits, $n$, calculated using trial Gaussian distributions.

Figure 4

Squeezed states generated on superconducting qubits. (a) A quantum circuit implementing a single QAOA layer for the MaxCut problem on a six-qubit system, producing a state with a reduced variance along the $z$ axis. (b) Squeezing measured with the quantum circuit in (a). The dashed lines show a noiseless qasm simulation. (c) The $p_m$ distribution of the four-qubit state with $-5.96\mathrm{dB}$ squeezing generated by a depth-two QAOA with optimal parameters ${\gamma }_1=0.918, {\gamma }_2=-0.257, {\beta }_1=-0.711$, and ${\beta }_2=-2.175$. The gray histogram shows the state ${\left(H|0\rangle \right)}^{\otimes n}$.

Figure 5

Entanglement from squeezing and quantum Fisher information. (a) The values of $\langle {\widehat{L}}_z^2\rangle$ (E1) obtained in simulation and hardware are close to zero, indicating that the states are in the vicinity of entangled Dicke states. (b) Number of entangled particles $k$ calculated from $F_Q\left[{\widehat{L}}_y\right]$ (E2) in simulation, and estimated for hardware using (E3).

Figure 6

Cross-resonance pulse schedules of the squeezing circuit and an arbitrary QUBO. (a) Quantum circuit of a general four-qubit fully connected cost operator $e^{-i\gamma {\widehat{H}}_C}$ transpiled to qubits 0, 1, 3, and 5 of ibm_lagos. (b) Pulse schedule of the cost operator used to generate the symmetric Dicke state, i.e., ${\omega }_{i,j}=1\forall i,j$. (c) Pulse schedule of a MaxCut instance with edge weights ${\omega }_{0,1}={\omega }_{0,2}={\omega }_{1,3}=-1$ and ${\omega }_{0,3}={\omega }_{1,2}={\omega }_{2,3}=1$. The circular arrows show where the phase shifts differ from the pulse schedule in (b). [(d) and (e)] MaxCut graph corresponding to the pulse schedules in (b) and (c), respectively. The duration of a single sample of the arbitrary waveform generators is $\mathrm{d}t=0.222\mathrm{ns}$. The light and dark pulses show the in-phase and quadrature of each complex amplitude pulse applied to control channels U0, U5, and U8 of ibm_lagos.

Figure 7

Squeezing as a function of the strength of gate noise modeled as bit-flip errors. A bit-flip error is added to every cnot gate by two-qubit Pauli channels, ${\mathcal{P}}_2={\mathcal{P}}_1\otimes {\mathcal{P}}_1$, where ${\mathcal{P}}_1=\sqrt{p_{\mathrm{error}}}X+\sqrt{1-p_{\mathrm{error}}}I$. The squeezing approaches zero as gate errors increase, showing the validity of squeezing as an application-tailored hardware benchmark.

Figure 8

Metrologically useful arbitrary Dicke states generated by a depth-three QAOA by minimizing the cost Hamiltonian in Eq. (14). The top panels show the Wigner quasiprobability distribution on the Bloch spheres. The bottom panels show the corresponding histograms of the total spin operator $\langle \widehat{Z}\rangle =m$. The orange numbers in each histogram show the overlap probability density $|\langle D_k^{12}|\psi \rangle {|}^2$ with the target Dicke states.

Figure 9

Advantage of QAOA initialized with squeezed states. The blue triangles show standard QAOA initialized from a coherent spin state. The orange circles and green stars show QAOA initialized with a spin-squeezed state created with $p_s=1$ and 2 QAOA layers, respectively. The $x$ axis is the QAOA depth after the initial state and the $y$ axis is the energy normalized to the ideal value. The markers and error bars indicate the average and variance of 100 graph instances drawn from $\mathcal{N}(4,0.5)$ with different sizes $n=4$ (top) to $n=12$ (bottom). Squeezed initial states boost the average energy of the QAOA optimized state as shown by the orange and green markers having an energy that is closer to the ideal energy than the blue markers. The energy increases only modestly as $p$ increases due to the complexity of the optimization landscape.

Figure 10

Multipartite entanglement from quantum Fisher information and number squeezing. (a) $F_Q$ witnessing $k$-partite entanglement for different $n$. (b) In the simulations, $F_Q$ obtained with ${\widehat{L}}_y$ is larger than ${\widehat{L}}_x$. The numbers of entangled particles ($k$) estimated from squeezing (${\sigma }_{\mathrm{css}}^2/\text{Var}\left({\widehat{L}}_{\text{z}}\right)={10}^{-\mathcal{S}/10})$ are close to the numbers obtained from $F_Q\left[{\widehat{L}}_y\right]$ for most cases. In a proper Dicke state as obtained with $n=4(p=2)$, the $\text{Var}\left({\widehat{L}}_{\text{z}}\right)$ becomes extremely small, leading to the very large value of ${\sigma }_{\mathrm{css}}^2/\text{Var}\left({\widehat{L}}_{\text{z}}\right)$ seen in the fourth row. (c) Illustrative examples of $k$-partite entanglement as entanglement depth.

Figure 11

Coupling map of ibmq_mumbai with the qubits used shown in violet. The four-, six-, and eight-qubit data were measured on the linearly connected qubits $\{19,22,25,26\}, \{14,16,19,22,25,26\}$, and $\{12,13,14,16,19,22,25,26\}$, respectively, chosen based on the cnot gate fidelity.

Figure 12

Illustration of the importance of alternating the cost function and the mixer operator. (a) A fragmented Wigner distribution on the Bloch sphere is obtained when ${\widehat{H}}_C$ is applied with $\gamma ={\gamma }_1+{\gamma }_2+{\gamma }_3$ from Fig. 1. (b) For the state in (a) no squeezing is observed at any $\beta$. The inset shows the probability distribution at $\beta =\pi /4$, corresponding to $\mathcal{S}=8\mathrm{dB}$, i.e., over-squeezing. (c) The energy landscape of the depth-one QAOA reveals that the lowest energy it can reach is $-35.47$, which is inferior to $-35.68$ obtained in depth-three QAOA.

Figure 13

Complexity of the optimization landscape for an $n=10$ vertices graph with edge weights [3.1, 3.5, 3.9, 3.6, 4.6, 4.2, 4.8, 3.8, 4.1, 4.8, 5.0, 4.8, 4.0, 3.8, 3.2, 4.1, 4.2, 4.6, 4.3, 3.5, 3.9, 3.8, 3.2, 3.2, 4.7, 3.7, 4.1, 3.5, 4.1, 4.0, 4.2, 3.6, 4.4, 4.1, 3.5, 4.2, 3.7, 3.4, 4.4, 4.4, 3.6, 4.0, 4.3, 4.9, 4.1] and QAOA depth $p=1$. Out of the full landscape, we show the first 0 to $2\pi$ portion of the $\gamma$ landscape in four subplots $\gamma \in [0,\pi /2],[\pi /2,\pi ],[\pi ,3\pi /2],[3\pi /2,2\pi ]$ with different color scales to increase the contrast between the local minima and maxima. This reveals a large number of local minima. The small improvement in solution quality with increasing $p$ can therefore be attributed to the many local minima, created from the interference of the frequencies generated by the different edge weights.