Direct implementation of a perceptron in superconducting circuit quantum hardware

The utility of classical neural networks as universal approximators suggests that their quantum analogues could play an important role in quantum generalizations of machine-learning methods. Inspired by the proposal in Torrontegui and García-Ripoll [Europhys. Lett. 125, 30004 (2019)], we demonstrate a superconducting qubit implementation of a controlled gate, which generalizes the action of a classical perceptron as the basic building block of a quantum neural network. In a two-qubit setup we show full control over the steepness of the perceptron activation function, the input weight and the bias by tuning the gate length, the coupling between the qubits, and the frequency of the applied drive, respectively. In its general form, the gate realizes a multiqubit entangling operation in a single step, whose decomposition into single- and two-qubit gates would require a number of gates that is exponential in the number of qubits. Its demonstrated direct implementation as perceptron in quantum hardware may therefore lead to more powerful quantum neural networks when combined with suitable additional standard gates.

Figure 1

(a) Schematic representation of a classical perceptron with $n=5$ inputs. The perceptron output $y$ is given by a nonlinear activation function $f(·)$ applied to the sum of input values $x_1,...,x_n$ individually weighted by $w_1,...,w_n$ and biased by a term $b$. A quantum analog (b) can be constructed, where the inputs are encoded in quantum states $|x_1\rangle ,...,|x_n\rangle$ of $n$ qubits and an output qubit undergoes a rotation by an angle of $\theta =2arcsin\left[c\right(x\left)\right]$ determined by the inputs values, leading to an excited population of ${\left|c\left(x\right)\right|}^2$ when starting the output qubit in the ground state.

Figure 2

(a) Diagram of the two-qubit experimental setup: two fixed frequency transmons are capacitively coupled via a flux-tunable coupler whose frequency determines the $ZZ$ coupling between the qubits. (b) Measured value of the $ZZ$ coupling as a function of the tunable coupler's frequency (orange dots). Here we plot the coupling strength $J$ conventionally defined via $H_{\mathrm{int}}=J|11\rangle \langle 11|$. In the perceptron context, the weight $w$ from Eq. (3) is given by $w=-J$. The solid line is the result of numerically diagonalizing the Hamiltonian of the system and the dashed line is the result of the lowest-order perturbative calculation (Appendix pp2).

Figure 3

(a, b) Illustration of the effect of a chirped pulse with initial and final frequencies ${\omega }_{\mathrm{i}}$ and ${\omega }_{\mathrm{f}}$ on a qubit with a frequency $\omega$. (a) If the initial and final detunings have the same sign, then the basis states $|0\rangle$ and $|1\rangle$ are unchanged (up to a phase). (b) If the signs are opposite, then the states get swapped. This results in a dependence of the final state on the shifted qubit frequency shown in panel (c). The perceptron activation function is realized by this dependence in the vicinity of $\omega \approx {\omega }_{\mathrm{f}}$ (darker line). (d) Time-dependence of the amplitude $\mathrm{\Omega }\left(t\right)$ and frequency ${\omega }_p\left(t\right)$ of an exemplary chirped pulse realizing the output-qubit rotation.

Figure 4

(a) The quantum perceptron activation functions: The excited state population $p_{\mathrm{e}}$ of the output qubit as a function of its final detuning $\mathrm{\Delta }$ after a chirped pulse is applied on the ground state of two-qubit system. Results for different pulse lengths $T$ are offset for visualization. The characteristic width of the activation function is inversely proportional to the pulse lengths $T$. The solid lines result from a simulation of unitary evolution, with no fit parameters in this model. (b) Frequency shift of the activation function dependent on the state of the input qubit, for a pulse length of $T=1.67\mu \mathrm{s}$ Both the magnitude and sign of this shift can be varied by tuning the frequency of the coupler. In these plots, the output qubit excitations are rescaled to correct for relaxation and readout imperfections, to highlight the agreement of the curve's shape with the simulation results. The frequency spacing between the simulated curves is not obtained by fitting but rather predicted from measured $J$ values [see Fig. 2]. The vertical lines indicate values of $b/2\pi =0.8\mathrm{MHz}$ (dashed) and $b/2\pi =4.0\mathrm{MHz}$ (dotted) which are used in Fig. 5.

Figure 5

(a) Final population of the output qubit as a function of the weight $w_1=-J$ for two different bias values $b/2\pi$ of $0.8\mathrm{MHz}$ and $4.0\mathrm{MHz}$. As expected, when the input qubit is in the $|0\rangle$ state, the final output state does not depend on the weight, while for control state $|1\rangle$ it follows the nonlinear activation function. The solid and dashed lines are derived from numerical simulations with no fit parameters. (b) Negativity of the final two-qubits state after preparing the input qubit in an equal superposition state and the output qubit in its ground state and applying the perceptron gate with a weight $w_1/2\pi =-5.2\mathrm{MHz}$. Negativity of a two-qubit state can range between 0 for a separable state to $1/2$ for a maximally entangled state. Lines show the simulated negativity for perfect unitary dynamics (solid) and when including relaxation of the qubits with decay times of $T_1=20\mu \mathrm{s}$ (dashed).

Figure 6

(a) A circuit that implements the equivalent of the perceptron gate with a single input and a single output qubit. A controlled-$V$ operation needs to be applied for each possible string of the inputs, in this case $x_{\text{in}}\in \{0,1\}$. (b) The circuit can be simplified by applying one of the multiqubit controlled operations as a single-qubit gate and adjusting the following operations. Here, $W(x=1)=V(x=1)·V{(x=0)}^{†}$. (c) The controlled-$W$ gate is further decomposed into at most two CNOT gates and three single qubit gates $A$, $B$, and $C$.

Figure 7

(a) Decomposition of the perceptron gate with two inputs and a single output qubit into controlled-controlled-$V$ operations and single-qubit operations $X$. (b) Improved circuit decomposition: one of the controlled-controlled operations, $V\left(00\right)$, is applied as a single-qubit gate with a simple adjustment $V\left(x\right)\to W\left(x\right)=V\left(x\right)V{\left(00\right)}^{†}$ for the other controlled-controlled operations. (c) Decomposition of a controlled-controlled-$W$ gate in terms of 6 CNOT gates and eight single-qubit gates.

Figure 8

Comparison for different gate times of the simulated populations, solid line, and fits to the analytical [Eq. (C2)] from Ref. [27]. Results for different pulse lengths T are offset for visualization. The analytical form well describes the simulated results, with small deviations for short gate times.