Virtual $Z$ gates and symmetric gate compilation

VZ gates eliminate the need for performing physical rotations about the Bloch $z$-axis, allowing us to focus solely on rotations in the $(x,y)$ plane. We denote by $R_{\phi}(\theta)$ a rotation by an angle $\theta$ about an arbitrary axis in the $(x,y)$ plane, making an angle $\phi$ with the $x$-axis:

We also denote

The physical implementation of $R_{\phi}(\theta)$ involves applying an on-resonance microwave pulse of the form $\varepsilon_{\mathrm{tot}}(t)=\varepsilon(t)\cos(\omega t+\phi)$ to the qubit, where the integrated pulse amplitude (for a given pulse duration) determines $\theta$, and the pulse phase determines $\phi$. The phase $\phi$ is arbitrary, as is the choice of the $(x,y)$ coordinate system, both set by the initial pulse. This illustrates how the VZ gate is implemented simply by updating the definition of which pulse phase corresponds to $\phi=0$ (usually set to be the $x$-axis, as above). However, this adjustment has tangible physical effects on subsequent gates: after a virtual $R_{z}(\varphi)\equiv\exp[-i(\varphi/2)\sigma_{z}]$ gate [note that $R_{z}(\pi)\equiv Z=-i\sigma^{z}$], the phase of each of the rotations that follow is shifted by $\varphi$. For example, when $\varphi=\pi$, then the next operation $R_{x}(\theta)$ becomes $R_{\pi}(\theta)=R_{-x}(\theta)$, i.e., a rotation about the $x$-axis becomes a rotation about the $-x$ axis, in the sense that $R_{-x}(\theta)=R^{\dagger}_{z}(\pi)R_{x}(\theta)R_{z}(\pi)$.

For most commercial cloud-based quantum processors not all rotations are natively available. For example, for the IBMQ devices, the calibrated single-qubit native gate set typically consists of the operations $\mathcal{G}=\{R_{z}(\varphi),\sqrt{X},X\}$, where $\sqrt{X}\equiv R_{x}(\pi/2)$ and $X\equiv R_{x}(\pi)=-i\sigma_{x}$, which are generated using $H(t)$ given in Eq. 1. However, these are not the only Clifford operations necessary for universal quantum computation. All other Clifford gates must be decomposed into these operations. Specifically, a $Y\equiv R_{y}(\pi)=-i\sigma_{y}$ gate requires an $X$ gate combined with VZ gates, which can be done in different ways. One method of compiling a $Y$ gate is asymmetrical:

Alternatively, a symmetric compilation of the $Y$ gate with respect to the VZ gates is:

Although these methods are theoretically equivalent in the sense that $Y^{\text{asym}}=Y^{\text{sym}}=Y$ is a mathematical identity, this is no longer the case when one accounts for deviations from unitary dynamics due to open quantum system effects, as we discuss in detail below.

To demonstrate how the two compilation strategies result in different outcomes, we consider a simple experiment, in which we apply sequences with a varying number of $YY$ pulses to the two orthogonal initial states $\ket{\pm i}$. Ideally, the fidelity of $YY$ applied to $\ket{+i}$ or $\ket{-i}$ should be identical. However, with the asymmetric decomposition the two states follow different Bloch sphere trajectories and leave the $(x,y)$ plane. That is, in the case of $Y^{\text{asym}}$, the virtual $R_{z}(-\pi)$ gate instantaneously interchanges $\ket{+i}$ and $\ket{-i}$ (up to a global phase of $i$) before the physical $X$ gate is applied. This has the effect of $\ket{-i}$ following a trajectory through the stable ground state $\ket{0}$ during the $X$ gate, while $\ket{+i}$ passes through the unstable excited state $\ket{1}$ [see Fig. 1(a)]. The second $Y$ gate leads to a reversal of this trajectory, again passing through the ground/excited, as the virtual $R_{z}(-\pi)$ reverses the direction of rotation. Consequently, $\ket{-i}$ experiences a lower relaxation rate and maintains a higher fidelity compared to $\ket{+i}$ over the course of repeated applications of the $YY$ sequence.

Conversely, using the symmetric decomposition $Y^{\text{sym}}$, the first VZ gate, $R_{z}(-\pi/2)$ transforms $\ket{-i}$ to $\ket{-}$ and $\ket{+i}$ to $\ket{+}$ (up to a global phase of $e^{i\pi/4}$). These states then undergo an $X$ gate, which leaves them unchanged (up to a global phase). The next VZ gate, $R_{z}(\pi/2)$, transforms the state back to its original position on the $y$-axis. Therefore, with this compilation, $\ket{\pm i}$ both remain in the $(x,y)$ plane at all times during the $Y^{\text{sym}}$ gate, and do not experience different relaxation rates. As a result, the fidelities of $\ket{\pm i}$ should be similar under $YY$, as for a physical $Y$ gate. By linearity, this extends to any state in the $(x,y)$ plane, i.e., to any superposition of $\ket{\pm i}$ or $\ket{\pm}$. Another way to see this result is that symmetric $YY$ compiles to two repetitions of Eq. 6, which is then equal to $R_{z}(\pi/2)XXR_{z}(-\pi/2)$. The interior $XX$ sequence traces out a full $2\pi$ rotation and thus always leads to trajectories that are symmetric about the $(x,y)$ plane, as expected by the $YY$ sequence.

We verified the effects predicted above through various experiments, using both the ibm_sherbrooke and MUNINN processor. As described above, we first prepare the initial states $\ket{\pm i}$, apply a series of $YY$ sequences, unprepare the initial state, and measure the system in the $\sigma_{z}$ eigenbasis. We define the empirical fidelity as the frequency of favorable outcomes, i.e., the number of $\ket{0}$ outcomes divided by the total number of experimental shots ($800$). Fig. 1(b) shows the results on the MUNINN processor, where we demonstrate that the asymmetric compilation of the $Y$ gate leads to the predicted asymmetry in the fidelity of the $\ket{\pm i}$ states. In contrast, the two states decay almost identically when the symmetric compilation of $Y$ gate is used. We observe the same effect after repeating the experiments on different qubits of the ibm_sherbrooke processor, as highlighted in Fig. 2.

Next, we consider the impact of asymmetric compilation on DD sequences. Specifically, we consider asymmetric and symmetric versions of the XY4 sequence [50]:

where $f_{\tau}=e^{-i\tau H}$ denotes the free evolution unitary generated by the total system-bath Hamiltonian $H$. We also consider how to correctly implement the $\overline{X}$ gate.

To test our prediction that when using $Y^{\text{asym}}$, XY4 is effectively the same as UR₄, Fig. 3 presents the results of measuring the fidelity of the $\ket{+}$ state as a function of time for a variety of different pulse sequences, each of which is applied repeatedly. Specifically, we apply the following sequences to a single qubit on ibm_sherbrooke: UR₄ using the symmetric definition for $\overline{X}$ given in Eq. 15, and two versions of XY4 using the symmetric and asymmetric $Y$ gates. As shown in Fig. 3, the UR₄ and XY4${}^{\text{asym}}$ sequences are almost indistinguishable, as expected. In contrast, XY4${}^{\text{sym}}$ is distinct.

It is important to note that Qiskit [51] natively compiles the $Y$ gate in the asymmetric form of Eq. 5. Therefore, caution is necessary when interpreting previously reported DD results involving transmon qubits that did not use the $Y^{\text{sym}}$ gate, including numerous studies involving the XY4 sequence.

Fig. 3 also displays the $YY$ and $X\overline{X}$ sequences, constructed using the symmetric definitions given in Eqs. 6 and 15, respectively. An unexpected feature observed in Fig. 3 is that all five sequences shown (including the robust ones), exhibit oscillations, which typically arise from coherent errors. We hypothesize that this phenomenon is due to an interference effect between consecutive pulses, e.g., due to an impedance mismatch in the microwave control lines [52].

To test this hypothesis, we repeated the same experiments as shown in Fig. 3 (except that we did not repeat XY4${}^{\text{asym}}$ since we already established its equivalence with UR₄), but with an intentional delay added between consecutive pulses, thus doubling the pulse interval $\tau=56.8$ ns, defined as the time delay between the peaks of two consecutive pulses. The result, shown in Fig. 4, is that the XY4${}^{\text{sym}}$ and UR₄ sequences no longer oscillate, but exhibit simple decay. Moreover, the stark difference between the latter two sequences seen in Fig. 3 has now almost disappeared. This is consistent with the observation that $Z$ (dephasing) errors are the dominant error source in transmon qubits, so that sequences suppressing $X$ or $Y$ errors have little added benefit over sequences suppressing only $Z$ errors.

The fact that the fidelities seen in Fig. 3 are higher for intervals of $2\tau$ rather than $\tau$ also helps to explain why previous studies involving transmon qubits [33, 34] have observed that, in contrast to the DD theory for ideal, zero width pulses (e.g., Ref. [53]), the optimal pulse interval is not always the shortest possible (the same phenomenon has also been observed in other platforms, e.g., nuclear magnetic resonance [54] and trapped ions [55]). The pulse interference effect, with pulses applied consecutively with the minimum shortest possible pulse interval $\tau$, can introduce additional coherent errors that result in inferior DD performance even with sequences (such as UR₄) that are robust against small coherent errors. Our results confirm the conclusion of Ref. [34] that it is essential to optimize the pulse interval for a given quantum processor, with the added insight that this optimization can reduce or (depending on the pulse sequence) even eliminate coherent errors due to pulse interference.

The reason we include the $X\overline{X}$ and $YY$ sequences in Fig. 3 is that $X\overline{X}$ is susceptible to phase errors, while $YY$ is susceptible to rotation errors [Eq. 2], as discussed in detail in Ref. [49]. More specifically, Figs. 3 and 4 shows that the two sequences exhibit oscillations for both the $\tau$ and $2\tau$ cases, with a period significantly shorter than that of the other sequences shown. This is consistent with the existence of single-pulse phase and rotation errors in addition to pulse interference errors. Doubling the pulse interval significantly increases the oscillation period, as seen in Fig. 4, but does not eliminate the oscillations. Moreover, we have checked (not shown) that further increasing the pulse interval to $3\tau$ has little effect on the $X\overline{X}$ and $YY$ fidelities, showing that coherent phase and rotation errors cannot be eliminated by controlling the pulse interference effect alone.

Both Figs. 3 and 4 display results from a single qubit. To test whether the small difference between the robust XY4${}^{\text{sym}}$ and UR₄ sequences [32] seen in Fig. 4 even with a doubled pulse interval is a statistically significant feature, we performed the XY4${}^{\text{sym}}$ and UR₄ experiments on all $127$ qubits of the ibm_sherbrooke device, for pulse intervals of $\tau=56.8$ns, $2\tau$, and $3\tau$. The results are shown in Fig. 5, after removing four of the qubits whose measurements were inadvertently performed during a calibration cycle (qubits 20, 21, 56, 63). We find that the oscillations exhibited by both XY4${}^{\text{sym}}$ and UR₄ in the $1\tau$ case entirely disappear in $122$ out of the $123$ qubits (the only exception being qubit $67$) for pulse intervals of $2\tau$ and $3\tau$. After fitting the fidelities to $a+be^{-t/T_{D}}$, a small difference in the decay constants remains for $2\tau$ intervals: $T_{D}(\text{UR}_{4})\geq T_{D}(\text{XY4}^{\text{sym}})$ in 59% of the cases. This difference disappears almost entirely for the $3\tau$ intervals: $T_{D}(\text{UR}_{4})\geq T_{D}(\text{XY4}^{\text{sym}})$ in 51% of the cases.

We may thus conclude that the pulse interference effect is significant when pulses are applied back-to-back but strongly diminishes when the pulse interval is doubled, and essentially disappears entirely when it is tripled.