We consider the two-level Hilbert space of a qubit, i.e., spun by states, |0⟩, |1⟩. We assume a θ-dependent quantum operation (gate) of a general form (Eq. 1): S θ = Z θ † S Z θ , (1) with the angle-dependent rotation (phase shift) around the z-axis on the Bloch sphere [17] (Eq. 2), Z θ = e − i θ / 2 0 0 e i θ / 2 , (2) written in the basis |0⟩, |1⟩. It is motivated by the implementation of qubit gates, e.g., in the IBM quantum computer, as explained in the following.

The operation S _θ can be applied n times, resulting in the total operation S θ n S acting on the initial ground state |0⟩. The sequence of operations is depicted schematically in Figure 1, with the details of pulse shapes, including the effect of the phase shift, shown in Figure 2.

We perform a dichotomic diagonal measurement M = α|0⟩⟨0| + β|1⟩⟨1| on the prepared state ρ, which gives the mutually exclusive outcomes 1, 0. A general form for the probability of 1, given some initial ρ and some given S, is (Eq. 3): p n θ = T r M S θ n ρ S θ n † = A n ⁡ sin ⁡ θ + B n ⁡ cos ⁡ θ + C n , (3) with some constants A _n , B _n , C _n , which are independent of θ. This is the simplest possible test of the phase shift to run on a generic quantum computer. Due to linearity, even changes in the gate S (e.g., due to a different calibration) do not affect the test’s validity. Only the A, B, and C constants will change.

Of course, this prediction will no longer be valid if (a) the actual Hilbert state is larger, with, e.g., an additional state |2⟩, (b) M, S, or ρ depend on θ, and (c) M is not diagonal.

Only such effects can lead to deviations from (3). For (c), only a second harmonics occurs. We discuss these possibilities in detail in Section 3, focusing on potential perturbative corrections.

The operations we use conform to native gates [18] provided by IBM quantum cloud computing, realized by microwave pulses. The perfect operation corresponding to our particular choice is given by M = |1⟩⟨1|, ρ = S|0⟩⟨0|S ^†, and S ≡ R X ( π 2 ) ≡ X . We intend to use this native gate as the simplest choice, but the actual gate can be different, over/under-rotated, or even a completely different one. It should be noted that S is just a π/2 rotation around the x-axis on the Bloch sphere [17] (Eq. 4). S = 1 2 1 − i − i 1 . (4)

An ideal π/2 rotation has eigenvalues ± i and 1, and the probability reads as follows (Eq. 5): p n θ = 1 − cos ⁡ θ / 2  for  n ≡ 1 m o d 4 1 / 2  for  n ≡ 0 m o d 2 1 + cos ⁡ θ / 2  for  n ≡ 3 m o d 4 . (5)

We test (A) the fit from (3) against the measured outcome of the preparation sequence for a specific n and (B) if p ₁ − p ₅ = 0 and p ₁ − 2p ₅ + p ₉ = 0 for an arbitrary θ. The rotation Z _θ is virtual and is performed together with S [12], resulting in a single operation S _θ . The actual effect of the Z gate is the rotation of in-phase and out-of-phase pulse components, not the gate pulse itself. Z gates (phase shifts) are a standard method to parameterize the native gates in the easiest possible way. The drive (interlevel) frequency is, in practice, much higher than the inverse time scale of the gate pulse. Therefore, the application of the Z gate reduces to a phase shift of the original gate pulse with the driving microwave field. It remains independent of the shape of the pulse as long as the shift is correct. It should be noted that an ideal gate S gives A = 0, but we do not have to assume it. On the contrary, keeping all three fit parameters, our test is robust against global phase shifts and decoherence. The great advantage of these tests is the usage of only a single qubit, partial independence of unknown properties of the environment and quantum operations, and universality. It applies to any two-level system. In practical implementation, it is helpful to eliminate memory effects of its repetitions by picking a random θ from a range uniformly covering any interval of length 2π. The result of the test should not change if adding a definite number of operations, S, at the end. It should be noted that in the ideal case, an even number of Ss after Z _θ gives the probability 1/2, while for an odd case, the probabilities of 0 and 1 get swapped every two Ss.

The IBM Quantum Platform cloud computing offers several devices, working as a collection of qubits, which can be manipulated by gates from a limited set—either single-qubit or two-qubit operations. The rotation Z is not a real but an instantaneous virtual gate VZ(θ) added to the next gate [12]. In particular, the sequence of gates Z _θ and S, shown in Figure 1, is realized by the native S _θ .

The names of the devices we tested are Armonk, Lagos, Lima, Jakarta, Perth, Nairobi, and Bogota. The collected statistics depend on the number of jobs, circuits, and shots, see Supplementary Appendix SA1. The linearity of the fit (3) makes the test robust against calibrations. Probabilities from different jobs can be simply averages, and so do the fitting parameters. The fitting has been performed by least squares. We compared the outcome statistics, the measured rate of occurred value 1, with the fit to (3). We also ran the circuits consisting of two and three gates S _θ instead of one to compare the possible deviations. For a benchmark, to estimate an error per gate, we used up to 70 S _θ . We made our scripts and collected data publicly available [19].

We present the results in Figures 3, 4. The standard deviation is calculated for the independent Bernoulli statistics, i.e., p ( 1 − p ) / N , where N is the number of repetitions (jobs, times, shots, times, and circuits per angle). In all runs, the prediction (3) is verified down to level 10^–3 of relative error. However, the deviations of order 10^–3 are significant when compared to the predicted error (more than 5 standard deviations). The deviation is smallest on Armonk—the single-qubit device—but still significant. It should be noted that the execution time of S or S _θ gates is 35.5 ns except on Armonk, where it is 71.1 ns. As similar results, exhibiting systematic θ-dependent deviations, have been obtained on different devices at different times (the data collection took 1 year, 2021–2022, while each run took from a few days to a few weeks), they deserve some physical explanation. The results from Bogota are consistent but indicate a noticeable phase shift; we discuss them separately in Supplementary Appendix SA2. The natural reason is that the actual performance of the gates can differ from the ideal expectations. For instance, a nonlinearity of the waveform generator can modify the pulse in a θ-dependent way [13].

We analyze a possible explanation of the deviations by non-ideal execution of the gates under the following four assumptions:

1. The two-dimensional Hilbert space of the quantum system

The standard realization of the gate S, including generic deviations in the actual pulse, can be described as (Eq. 6): S θ = T ⁡ exp ⁡ ∫ 0 π / 2 e i θ 1 + ϵ 0 1 + e − i θ 1 + ϵ * 1 0 d ϕ / 2 i (6) for some complex ϵ(θ, ϕ) = ϵ _r + iϵ _i describing the parameter-dependent imperfection of the gate, including over/under-rotations and other coherent errors, where T denotes the chronological product in the Taylor expansion of the exponential of the integral with respect to ϕ corresponding to dimensionless gate operation time. The gate is ideal for ϵ = 0. We consider only off-diagonal corrections because the pulse only modulates the driving frequency of the transition between levels. We find the first-order correction to the probability (3) for n gates S _θ in the form (see Supplementary Appendix SA3) δ p n θ = ∫ 0 π / 2 d ϕ × − sin ⁡ θ ⁡ sin ⁡ ϕ ϵ i / 2  for  n ≡ 1 m o d 4 sin ⁡ θ cos ⁡ ϕ − sin ⁡ ϕ ϵ i / 2 − cos ⁡ θ ϵ r n / 2  for  n ≡ 2 m o d 4 sin ⁡ θ ⁡ cos ⁡ ϕ ϵ i / 2  for  n ≡ 3 m o d 4 cos ⁡ θ ϵ r n / 2  for  n ≡ 0 m o d 4 . (7)

At first sight, the model above could, in principle, explain the deviations because most of the deviations cross 0 at θ = 0, π in Figures 3, 4 (taking into account a general shift of the angle on Bogota) and the deviations for a single and three S _θ sum approximately to zero if the symmetry ϵ(ϕ) = ϵ(π/2 − ϕ) is assumed. Nevertheless, this must be confirmed by tracing down to the actual pulse formation. The results for S θ 2 presented in Figure 4 are not fully compatible with this model at θ = +π/2, but it may be the result of the assumption of identical subsequent gates which may not be fully realized.

To fully test the model in Eq. 7, we re-run the tests to compare the results from 1 to 5 gates S _θ , which should give identical deviations, according to the model, i.e., p ₁ − p ₅ = 0 in the first order. We performed the tests on Lagos (qubits 0 and 6), Lima (qubit 0), and Perth (qubit 0), running with shuffled angles to avoid memory effects. The results show extraordinary deviations for Lagos qubit 0 in the first run in February 2022 (however, we found such large deviations already in November 2021 in the benchmark test), but repeating the test in May 2022 gave much smaller deviations, see Figure 5. The deviation from the first run is large, ∼ 1 0 − 2 , and gets inverted between 1 and 5 gates. We additionally checked that the inversion took place in each of the four gates in the benchmark test. Here, the reason must have been completely different, e.g., a considerable technical or fundamental problem. Such a large deviation can be explained by an enlarged Hilbert space, including the multiplication of quantum states, in analogy to the many-copies idea [15, 16], but this needs further analysis to confirm or rule out. Smaller deviations in the second run indicate that they may depend on calibration, which is applied to the qubit daily, although they stayed on the same level for 6 months, ruling out the effect of an incidental calibration. The qubit 6 from Lagos and qubits 0 from Lima and Perth also give much smaller deviations, Figures 6, 7, although still they do not match for 1 and 5 gates, small but still nonzero (beyond 5 standard deviations) differences.

Standard models of decoherence do not depend on gate parameters (θ in our work). Benchmarking the decoherence rate at ∼ 1 0 − 3 , comparable to the observed deviations, eliminates it as the main reason, see Supplementary Appendix SA4. The relaxation and decoherence times are of the order of at least ∼ 10 μ s [20], which is far beyond the gate time of 35 ns. In Supplementary Appendix SA5, we present an analytical argument for such a model, including readout error, relaxation, depolarization, and phase damping [17], to show that p ₅ ≃ p ₁ still holds up to first-order corrections. The magnitude of the statistical errors compares favorably with the real device error estimates, such as in Figure 6. Finally, using sequence or up to 9 gates S _θ , Figure 1, we checked if p ₁ − 2p ₅ + p ₉ = 0 on Nairobi (January–February 2023), which should hold up to second-order deviations, see Supplementary Appendix SA5. The result is beyond 4 standard deviations, see Figure 8. We conclude that the real device deviation from a test value, which we observe, goes beyond the standard models of noise and the known amount of leakage [21].

A higher state [14] as an alternative explanation seems unlikely due to different transition frequencies and the fact that it is a second-order correction (see the analysis in Supplementary Appendix SA6). A simple in-phase/quadrature (I/Q) imbalance [12] cannot explain the dominating second harmonic in the deviations, as it would give only third harmonics (see Supplementary Appendix SA7). An even more complicated description, like a considerable extension of the Hilbert space [15, 16], would be the last option.