*By Moein Malekakhlagh IBM Quantum, IBM Thomas J. Watson Research center*

Pushing for higher quantum volume, and reaching the thresholds necessary for error correction, requires extremely precise elementary single- and two-qubit gates. These gates are the quantum analogue of elementary operations in classical computation. For instance, NOT is a classical single-bit operation through which the logical state is flipped, while XOR is a two-bit operation, which essentially adds the inputs modulo 2 (so, 0 + 1 = 1, but 1 + 1 = 0). However, in quantum systems, we can also perform rotations on the quantum state of the qubits. For instance, if we think of a qubit’s possible states as sitting on a sphere, where |0> sits at the top and |1> sits at the bottom, then the quantum NOT gate corresponds to a rotation around the X axis of the qubit. Analogously, the Controlled-NOT (CNOT) gate, quantum analog of the XOR, is a two-qubit gate in which the target qubit goes through a rotation around its X axis only when the control is in logical state |1> (see figure 1).

Cross-resonance (CR) [1, 2] is a microwave-activated two-qubit gate performed by driving one of the qubits (control) at the frequency of the other (target) (see figure 2). The qubits are connected through effective capacitors in a superconducting circuit. The CR drive protocol produces entanglement between the qubits, mainly through an interaction of the form ZX—rotation of the two-qubit wavefunction around the Z-X axis—with which a CNOT gate can be calibrated. Simplicity in implementation, resilience to noise and acceptable overhead has made architectures based on CR gates with fixed-frequency transmon qubits a favorable choice for current IBM processors.

Optimizing the CR gate fidelity requires an accurate understanding of the underlying interactions as well as the ability for precise control. Initial proposals [1, 2] considered a two energy level model for the qubits. This assumption captures only the dominant two-qubit interactions, such as the ZX rate and the drive-induced frequency shift on the control qubit known as Stark shift. Since the CR gate is implemented using weakly anharmonic transmon qubits, there are additional unwanted interactions due to higher quantum states of the qubits (figure 2). Therefore, more accurate multi-level modeling of the aforementioned error terms is necessary [3, 4, 5].

There are two main approaches towards understanding involved time-dependent quantum interactions. First, we could attempt to do precise numerical solution of the dynamics. This is essential in making reliable comparisons with experiment. However, it is typically not straightforward to gain immediate intuition about the nature of this system’s underlying interactions. In our recent paper [5], we used a perturbative approach, called Schrieffer-Wolff perturbation theory (SWPT), to solve for an effective gate Hamiltonian starting from a multi-level physical model for the system. Perturbation here means that we add the effective interactions, in this case due to the CR drive, as a small addition to the static Hamiltonian of the qubits.

In the SWPT method, we take an average of the off-resonant high-frequency states, arriving at effective low-frequency models of these interactions for the system. For the CR gate, this means that the impact of all higher qubit states is encoded in the effective two-qubit interaction rates. Contrary to rotating-wave approximation, which discards off-resonant interactions, SWPT accounts for their effect by solving for a series of perturbative frame transformations. The choice of effective frame is set by the drive scheme for the gate. For the CR gate, where the drive is resonant with the target qubit frequency, the effective frame is block-diagonal (a matrix whose diagonal elements are square matrices, and all other elements are 0) with respect to the control qubit, resulting in an effective Hamiltonian of the form: [3, 5]

$$\hat{H}_{eff} = \omega_{ix} \frac{\hat{I}\hat{X}}{2} + \omega_{iz} \frac{\hat{I}\hat{Z}}{2} + \omega_{zi} \frac{\hat{Z}\hat{I}}{2} + \omega_{zx} \frac{\hat{Z}\hat{X}}{2} + \omega_{zz} \frac{\hat{Z}\hat{Z}}{2}. \quad (1) $$

The CNOT gate is typically calibrated using the ZX and IX rates, while the rest result in coherent error for the gate. State of the art CR implementation using transmon qubits can approximately result in 1 MHz of IX, 2 MHz of ZX, 10 MHz of ZI, 20 KHz of IZ and 100 KHz of ZZ interactions (see figure 3).

The effective gate parameters in Eq. (1) depend on qubit-qubit frequency difference (detuning) \(\Delta_{ct}\), qubit anharmonicities \(\alpha_c,\alpha_t\) (that is, the anharmonicity of the control and of the target, respectively), exchange coupling \(J\) and drive amplitude \(\Omega\) (see figure 2). For instance, the leading order desired ZX and parasitic ZZ rates are found as:

$$ \omega^{(1)}_{ZX} = \frac{J\Omega}{\Delta_{ct} + \alpha_c} - \frac{J\Omega}{\Delta_{ct}}, \quad (2)$$

$$ \omega^{(0)}_{ZZ} = \frac{J^2}{\Delta_{ct} - \alpha_t} - \frac{J^2}{\Delta_{ct} + \alpha_c}. \quad (3)$$

According to Eqs. (2) and (3), effective CR rates are very sensitive to the ratio of qubit-qubit detuning and anharmonicity and various regions of operation emerge [4, 5] (see figure 3). To maximize the intended ZX rate, the detuning must be in regions II and III of the graph below—close to the positive side of the straddling regime 0 < \(\Delta_{ct}\) < \(- \alpha_{c}\) . The unwanted ZZ rate, however, is maximized close to the resonance with the third level of each qubit, i.e. \(\Delta_{ct} = \alpha_c\), and \(\Delta_{ct} = -\alpha_t\), and suppressed in the middle of regions I and II.

Figure 3 (from Ref. [5]): ZX and ZZ rates as a function of drive Ω and qubit-qubit detuning Δ_ct. The effective rates are calculated using SWPT up to fourth order in drive amplitude Ω. |

The average gate error is what typically governs the performance of any gate, and we can estimate it experimentally using randomized benchmarking (RB) [6, 7]. Generally, to optimize for average CR error, one needs to maximize the ZX (IX) rate and minimize other unwanted interactions in Eq. (1). One strategy is to employ an echo pulse sequence as shown in figure 3 [8, 9]. It consists of two CR pulses with opposite amplitude, accompanied with \(\pi\) rotations on the control qubit. The echo creates constructive/destructive interference between the effective gate parameters. Through this procedure, it removes unwanted rates such as ZI, ZZ and IX, at the expense of inducing higher order error terms. We find that optimal choices for qubit-qubit detuning are achieved in the middle of regions II, I and III, consistent with the behavior of ZX and ZZ rates in figure 4. For typical qubit anharmonicity of -330 MHz, these sweet spots are approximately centered around -100, 100 and 200 MHz, respectively.

Figure 4 (from Ref. [5]): An echo sequence consisting of two CR tones with flipped drive amplitude with two π pulses on the control qubit [8, 9] and the resulting average gate error. |

An alternative gate calibration based on the CR architecture is to tune a direct CNOT gate without echoing [10, 11]. To achieve this, we want no action on the target qubit when the control is in state |0> and a \(\pi\) rotation on the target when the control is in state |1> . For this calibration, the IX rate is not an error term anymore, and the gate speed is determined by IX-ZX rates resulting in a faster operation. Equipped with interference couplers to suppress the static ZZ rate, and using a virtual frame change to cancel out Stark shifts, Ref. [10] demonstrates a 180 ns gate with 99.77% gate fidelity.

In a quantum processor, CR pairs are connected to neighboring spectator qubits that can influence the gate operation due to frequency crowding—the operating frequencies between neighboring qubits being close enough to cause unwanted interactions. This poses further restrictions on the optimization of both two-qubit gate fidelity as well as multi-qubit interactions that affect holistic measures such as quantum volume [12]. An immediate three-qubit extension of the CR gate involves a third qubit that is either coupled to the control (control spectator) or the target qubit (target spectator). Applying SWPT on these toy problems reveal a series of two- and three-qubit frequency collisions (resonances) that need to be avoided in processor design (see figure 5).

In conclusion, CR offers a very promising platform for two-qubit gate implementation. State of the art calibrations of the CR gate using IBM quantum processors can reach 99.40% fidelity with overall gate time of approximately 280 ns for an echo sequence [10], and 99.78% fidelity with an overall gate time of 180 ns for a direct CNOT with interference couplers [11]. Further improvement of the gate fidelity requires a continual interplay between theory and experiment.

Acknowledgements: Optimal control techniques for CR gate has been a long-standing research and engineering effort among the IBM quantum team, for which we appreciate helpful discussions with and acknowledge earlier works by Easwar Magesan, Emily Pritchett, David C McKay, Sarah Sheldon, Ken Wei, Abhinav Kandala, Neereja Sundaresan, Peter Jurcevic, Isaac Lauer, Jerry M Chow, and Jay M Gambetta.

References:

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