Demonstrating a Continuous Set of Two-Qubit Gates for Near-Term Quantum Algorithms

Quantum algorithms offer a dramatic speedup for computational problems in material science and chemistry. However, any near-term realizations of these algorithms will need to be optimized to fit within the finite resources offered by existing noisy hardware. Here, taking advantage of the adjustable coupling of gmon qubits, we demonstrate a continuous two-qubit gate set that can provide a threefold reduction in circuit depth as compared to a standard decomposition. We implement two gate families: an imaginary swap-like (iSWAP-like) gate to attain an arbitrary swap angle, $\theta$, and a controlled-phase gate that generates an arbitrary conditional phase, $\varphi$. Using one of each of these gates, we can perform an arbitrary two-qubit gate within the excitation-preserving subspace allowing for a complete implementation of the so-called Fermionic simulation (fSim) gate set. We benchmark the fidelity of the iSWAP-like and controlled-phase gate families as well as 525 other fSim gates spread evenly across the entire $\mathrm{fSim}(\theta ,\varphi )$ parameter space, achieving a purity-limited average two-qubit Pauli error of $3.8\times {10}^{-3}$ per fSim gate.

Figure 1

Demonstration of the tunable coupling between gmon qubits. (a) Pulse sequence used to measure swapping as a function of coupler bias. We initialize one qubit, perform an fSim gate defined by a set of three flux pulses that control the qubit frequencies and the coupling between the qubits, and measure the population of the other qubit. (b) We vary the fSim gate as a function of the length and amplitude of the coupler pulse to measure the swap rate as a function of the bias amplitude. (c) By taking the Fourier transform of the oscillations in (b), we extract the coupling strength, $|g|$, as a function of coupler bias. The coupling changes sign at the two “OFF” biases, ensuring we can turn the coupling off.

Figure 2

Exploring the parameter space of two-qubit gates. Each pixel represents one experiment. We use a set of 15 ns rectangular current bias waveforms to perform some fSim unitary by setting the qubit-qubit detuning, $\mathrm{\Delta }$, and the coupling strength, $g$. (a) To identify the low-leakage gates described by the fSim model, we measure leakage by initializing the $|11⟩$ state and measuring the $|02⟩$ state. When the detuning is near the qubit nonlinearity, we observe the expected Rabi oscillations. (b) We measure the conditional phase, $\varphi$, by performing a Ramsey experiment where we initialize one qubit with an $\mathrm{X}/2$ gate and perform tomography to measure the difference in accumulated phase ($\varphi$) with and without initializing the other qubit to the $|1⟩$ state. By choosing combinations of $\mathrm{\Delta }$ and $g$ as indicated by the CPHASE dash-dotted line [chosen as the low-leakage coupling strength from (a)], we are able to achieve any $\varphi$: [$-180°$, 180°]. (c) We measure the swap angle, $\theta$, by initializing the $|01⟩$ state and measuring the $|10⟩$ state. By placing the qubits on resonance and varying the coupling strength along the iSWAP-like dashed line, we are able to achieve any $\theta$: [0°, 90°].

Figure 3

Characterizing the iSWAP-like and CPHASE gate families with cross-entropy benchmarking. We plot the optimized fSim control angles, $\theta$ and $\varphi$, on the left y-axes and the Pauli gate error per two-qubit gate on the right y-axes, conservatively assuming $7.5\times {10}^{-4}$ single-qubit Pauli gate errors. (a) Characterization of the CPHASE gate family corresponding to $\text{fSim}(\theta \approx 0°,\varphi )$. Each gate is 15 ns long, consisting of control pulses that vary the qubit detuning, $\mathrm{\Delta }$, around the qubit nonlinearity, $\eta$, with a coupler bias amplitude chosen to complete one full swap: $|11⟩\to |02⟩\to |11⟩$. We measure an average two-qubit Pauli error of $1.9\times {10}^{-3}$ for the CPHASE family. (b) Characterization of the iSWAP-like gate family corresponding to $\text{fSim}(\theta ,\varphi \propto {\theta }^2)$. Each gate is 13 ns long, consisting of control pulses that place the qubits on resonance and vary the coupling strength, $|g|$, to achieve an arbitrary swap angle $\theta$ between the $|01⟩$ and $|10⟩$ states. We measure an average two-qubit Pauli error of $1.2\times {10}^{-3}$ for the iSWAP-like family.

Figure 4

Benchmarking the fSim gate set. Using XEB, we measure the Pauli error per cycle and subtract off a conservative estimate for the single-qubit Pauli gate errors of $7.5\times {10}^{-4}$. (a) We plot the two-qubit Pauli error for 525 fSim gates where $\theta$ and $\varphi$ have been constrained to a grid. (b) Histogram of both the error and purity loss for each gate presented in (a). Here we confirm a purity (coherence) limited average error for our fSim gates of $3.83\times {10}^{-3}$.