Following [99,27] and basic ideas fromChapter 2we can write the system Hamiltonian in the lab frame as,
H/¯h = 1 2ω1σ z 1+ 1 2ω2σ z 2 + Jσ1xσx2, (4.1) where σ{x,y,z}
i are the Pauli spin operators and the resonator energy has been ne-
glected. Throughout this chapter, Q2 will be used as the control qubit and Q1 will always be the target. The notation in places will be |Q2, Q1⟩. Similarly, Pauli op- erators indexed with 1 and 2 refer to operations on Q1 and Q2, respectively. J in Eq. (4.1) is the always-on resonator-mediated coupling strength between the two qubits. The analytic expression for J, neglecting higher levels of the transmon, is [74]
J = g1g2
2 (1/∆1 + 1/∆2), (4.2)
where gi is the coupling strength of qubit i to the resonator and ∆i is the detuning
between qubit i and the resonator frequency. This Hamiltonian can be diagonalized, resulting in a slight qubit frequency shift ˜ω{1,2} = ω{1,2}± J/∆, where ∆ = ω1− ω2 is
the detuning between the qubits. The key to the cross-resonance effect is a σx drive
Figure 4-1: Cross-resonance energy levels. Energy level diagram for the cross resonance interaction. The undressed energy levels for |01⟩ and |10⟩ are represented by dashed lines and the dressed states corresponding to ˜ω = ω ± J/∆ are indicated by the solid lines. With the control qubit in the |1⟩ state (red), a drive at ˜ω1 incident
on the control qubit induces rotations in the target qubit’s Bloch vector in a direction opposite to rotations induced when the control is in the |0⟩ state. This rate is recuded by a factor of J/∆ from a resonant dirve. Reproduced from Chow et al. [27].
as in Figure 4-1. To see this, we can write the Hamiltonian when qubit drives are present, H/¯h = 1 2ω1σ z 1 + Ω1cos(ωrf1 t + φ)σx1 + 1 2ω2σ z 2 + Ω2cos(ωrf2 t + φ)σ2x+ Jσ1xσ2x, (4.3)
where Ωi and ωirf/2π are the amplitude and frequencies of the drives on qubit i; ωi/2π
are the qubit ground-to-excited transition frequencies of qubit i and J/2π is the trans- verse exchange coupling strength in the absence of driving. Assuming the detuning between qubits is much larger than their linewidth and making the rotating wave approximation, this Hamiltonian can be taken through a series of unitary transfor- mations (see [99]) to highlight the nonlocal dynamics. The remaining terms oscillate rapidly unless certain drive conditions are met. Chief among these is ωrf
frame rotating at the qubit and drive frequencies, a static term develops [99] HDFeff/¯h = Ω1J 4∆ (cosφ1σ z 1σx2 + sinφ1σz1σ y 2). (4.4)
With a proper choice of phase φ and drive time, this term becomes a two-qubit inter- action and leads to unitary evolution as exp(−iπσz
1σ2x/4)
1. To shorten the notation, I will make the substitution σx = X, σ
y = Y, σz = Z along with the identity operation
I for the Pauli spin operators. A group of two Pauli operators like ZX is shorthand notation for ˆZ ˆX = σz ⊗ σx, where the first operator operates on the control qubit
subspace and the second operates on the target subspace.
In the previous analysis, we have ignored some complicating factors. Given the microwave environment on the chip, a level of crosstalk is unavoidable. A strong drive resonant with the target qubit frequency will leak onto the chip and be sensed by the target qubit independent of the state of the control qubit. In practice, this value is on the order of the drive felt by the target qubit. This leads to an IX term involving Rabi rotations of the target qubit that are independant of the state of the control qubit. A second complication comes from the control qubit being driven strongly off resonance. This leads to an ac-Stark shift ZI term [33] in the drive Hamiltonian. The effective Hamiltonian when the system is driven becomes
Hef f/¯h = ϵ(t)(ZI + (m − ν)IX + µZX), (4.5)
including the ac-Stark shift term, an IX crosstalk term and the CR term from (4.4). In the case of ideal two-level qubits, µ = J/∆. In general, the crosstalk term has a quantum and classical component. The ν parameter quantifies the amount of quantum crosstalk during the drive and m quantifies the classical component. In this chapter, I will forgo any analysis on the ν or m parameter and focus instead on µ and
1
More generally, the interaction can be written exp(−iβπσz
1σ2x/4) with a parameter β =
the effects of the higher energy states in the transmon. As noted elsewhere [33], these interactions do not degrade the conditional CR term ZX, as these terms commute.