Superadiabatic geometric quantum gates

The SAGQG proposal25 builds upon the concept of the Aharonov-Anandan type non-adiabatic geometric phase.29 For the Aharonov-Anandan phase to be solely of geometric nature, in the total phase

$${\mathrm{\Phi }} = - {\int}_0^T \langle \psi (t)|H(t)|\psi (t)\rangle dt + {\int}_0^T \langle \tilde \psi (t)|i(\partial /\partial t)\tilde \psi (t)\rangle dt$$ (1)

the first, dynamic-phase term must vanish (here \(|\tilde \psi (t)\rangle\) is the reference section state on the projective Hilbert space \({\cal P}\)30). This can be achieved by driving the state vector with a driving field that is applied perpendicularly to the state vector at all times. Under this condition driving the state vector on the Bloch sphere, the solid angle \({\tilde{\mathrm \Omega }}\) enclosed by the Bloch vector trajectory determines the acquired geometric phase\(\gamma = {\mathrm{\Phi }} = {\tilde{\mathrm \Omega }}/2\) (Fig. 1a).

Fig. 1 Superadiabtic geometric quantum gate concept. a Anticipated “orange slice” Bloch sphere trajectory (blue) enclosing the solid angle \({\tilde{\mathrm \Omega }} = 2\gamma\) (red). b Two-level system and microwave field parameter (detuning Δ(t), Rabi frequency Ω S (t) and phase φ + ϕ S (t)) utilized for the realization of superadiabatic geometric quantum computation Full size image

The Aharonov-Anandan phase is restricted to generate U(1) phase shift gates. The total Hamiltonian of the SAGQG is constructed employing the technique of transitionless driving26 where a reverse engineered correction Hamiltonian compensates for undesired transitions between the basis states. This way the effective superadiabatic Hamiltonian drives the instantaneous eigenstates exactly such that non-adiabatic correction terms are cancelled and the evolution of dynamic phases is fully suppressed, even within the fast driving regime.

Considering a two-level system with a time-dependent single driving field, our original Hamiltonian H 0 (t) has the following form in the co-rotating reference frame of the external driving field

$$H_0 (t)= \frac{\hbar }{2}\left( {\begin{array}{*{20}{c}} {{\mathrm{\Delta }}(t) + {\dot{\mathrm \Delta }}(t)t} & {{\mathrm{\Omega }}_{\mathrm{R}}(t)e^{ - i\varphi }} \\ {{\mathrm{\Omega }}_R(t)e^{i\varphi }} & { - ({\mathrm{\Delta }}(t) + {\dot{\mathrm \Delta }}(t)t)} \end{array}} \right),$$ (2)

where the driving field is applied with a detuning Δ(t) = ω 0 − ω D (t), with ω 0 the qubit resonance frequency and ω D (t) the driving field frequency, phase φ, and Rabi frequency Ω R (t). The non-standard form of the Hamiltonian in Eq. (2) in the rotating frame of the driving field arises from its time-dependent detuning (see Supplementary Information for details on the derivation of Eq. (2)). Exploiting the concept of TQD26 and deriving a suitable correction Hamiltonian H c Liang et al.25 propose the superadiabatic Hamiltonian

$$H_{\mathrm{S}}(t) = H_0 + H_c = \frac{\hbar }{2}\left( {\begin{array}{*{20}{c}} {{\mathrm{\Delta }}(t) + {\dot{\mathrm \Delta }}(t)t} & {{\mathrm{\Omega }}_{\mathrm{S}}(t)e^{ - i[\varphi + \phi _{\mathrm{S}}(t)]}} \\ {{\mathrm{\Omega }}_{\mathrm{S}}(t)e^{i[\varphi + \phi _{\mathrm{S}}(t)]}} & { - ({\mathrm{\Delta }}(t) + {\dot{\mathrm \Delta }}(t)t)} \end{array}} \right),$$ (3)

where \({\mathrm{\Omega }}_{\mathrm{S}}(t) = \sqrt {{\mathrm{\Omega }}_{\mathrm{R}}(t)^2 + {\mathrm{\Omega }}_{\mathrm{C}}(t)^2}\) is the superadiabatic Rabi frequency, and φ S (t) = arctan[Ω C (t)/Ω R (t)] is the superadiabatic phase. The corrected Rabi frequency is \({\mathrm{\Omega }}_{\mathrm{C}}(t) = \left[ {{\dot{\mathrm \Omega }}_{\mathrm{R}}(t)({\mathrm{\Delta }}(t) + {\dot{\mathrm \Delta }}(t)t) - {\mathrm{\Omega }}_R(t)\partial _t({\mathrm{\Delta }}(t) + {\dot{\mathrm \Delta }}(t)t)} \right]{\mathrm{/\Omega }}^2\), where the generalized Rabi frequency is introduced as \({\mathrm{\Omega }} = \sqrt {{\mathrm{\Omega }}_{\mathrm{R}}(t)^2 + ({\mathrm{\Delta }}(t) + {\dot{\mathrm \Delta }}(t)t)^2}\). The instantaneous eigenstates of the original Hamiltonian H 0 (t) are \(\left| {\lambda _ \pm (t)} \right\rangle\). (The explicit expression of the superadiabatic Rabi frequency Ω S (t), detuning \({\mathrm{\Delta }}_{\mathrm{S}}(t) = {\mathrm{\Delta }}(t) + {\dot{\mathrm \Delta }}(t)t\) and phase φ(t) are given in the 'Methods' section.)

In order to realize universal quantum computation the SAGQG applies a strategy previously developed by Zhu and Wang23,31 which is based on choosing a pair of orthogonal states \(\left| {\lambda _ \pm (t)} \right\rangle\) undergoing a cyclic evolution: \(\left| {\lambda _ \pm (T)} \right\rangle = {\mathrm{exp}}\left[ {i\phi _ \pm } \right]\left| {\lambda _ \pm (0)} \right\rangle\). Over the full length of the SAGQG transformation the dynamic phase is designed to cancel such that the system evolution becomes fully geometric \(U(T,0)\left| {\lambda _ \pm (0)} \right\rangle = {\mathrm{exp}}\left[ { \pm i\gamma } \right]\left| {\lambda _ \pm (0)} \right\rangle\) where the evolution operator U(T, 0) imprints only a U(1) phase factor on each of the eigenstates \(\left| {\lambda _ \pm (0)} \right\rangle\). Wang and Zhu ingeniously identified that these trivial phase factors on the \(\left| {\lambda _ \pm (0)} \right\rangle\) nevertheless translate to a non-Abelian transformation on the computational states in the co-rotating frame. Thus, even though the SAGQG is not based on a non-Abelian holonomy, in virtue of the elaborate basis transformation between the cyclic states and the computational states the U(1) geometric-phase factors convert to a non-Abelian, geometric transformation of the computational states allowing for universal quantum computation.

Bloch sphere trajectory

The SAGQG state evolution is based on a sequence of four trajectory segments of time duration τ, leading to a total gate length of t Gate = 4τ (see Methods section for details). We investigate and visualize the quantum gate Bloch sphere trajectory of a qubit initialized into the |0〉 state in a stroboscopic manner by applying projective readout pulses at times t m . As two representative gates we realize the Pauli-Z (Fig. 2a, b) and the Pauli-X (Fig. 2c, d) gate. The measured Bloch vector trajectories (dots) are in very good agreement with the numerically calculated trajectories (solid lines). Rotations around the y-axis (Pauli-Y gate) can be realized by setting φ = π/2 for the original Hamiltonian H 0 (t). In the realization of the Pauli-Z gate, the non-adiabatically obtained original trajectory is observed (compare Fig. 1a), since the input eigen state |0〉 is equivalent to the Hamiltonians instantaneous eigenstate \(\left| {\lambda _ \pm (0)} \right\rangle\) at t = 0. The particular shape of the trajectory in Fig. 2c, d illustrates that the geometric phase is obtained utilizing a sophisticated parameter time-dependence.

Fig. 2 Superadiabatic geometric gate realization: a Simulated and reconstructed Bloch sphere trajectory of the superadiabatic geometric Pauli-Z gate in the driving field frame for a spin initialized into the |0〉 state. b Bloch vector components u(t) = ρ 10 (t) + ρ 01 (t) (blue), v(t) = i(ρ 01 (t) − ρ 10 (t)) (orange) and w(t) = ρ 00 (t) − ρ 11 (t) (green), where ρ(t) is the density matrix representation of the instantaneous state, of the trajectory presented in a versus the gate time in multiples of τ. Solid lines represent numerically calculated trajectories and dots indicate measured values. Analogously c and d follow for the realized Pauli-X gate. e–g Measured population of the |0〉 state for a spin initialized into the states e |0〉, f \(1/\sqrt 2 \left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right)\) and g \(1/\sqrt 2 \left( {\left| 0 \right\rangle + i\left| 1 \right\rangle } \right)\) in dependence on γ for superadiabatic rotations around the x (green) and z-axis (blue). Dashed lines represent the expected values. Bloch spheres h, i, k indicate the initialized state (red arrow) Full size image

Generalization to geometric gate with arbitrary phase value

So far we demonstrated that rotations by γ = π/2 around the x and z-axis can be fulfilled with high fidelity by performing superadiabatic geometric quantum computation. In addition, by varying the opening angle of the “orange slice” trajectory an arbitrary geometric phase γ can be acquired. Utilizing the states |0〉, \(1/\sqrt 2 \left( {\left| 0 \right\rangle - \left| 1 \right\rangle } \right)\) and \(1/\sqrt 2 \left( {\left| 0 \right\rangle + i\left| 1 \right\rangle } \right)\) we demonstrate the rotation for different geometric phases γ (see Fig. 2e–k). In order to visualize the phase gate we map the acquired phase into a population by application of a projective π/2-pulse around the \(\bar y\)-axis. Hence, we show that the SAGQG concept additionally allows for the generation of an arbitrary phase shift gate. Collectively with the former we thus provide a universal set of single-qubit geometric quantum gates.

Fidelity assessment and fault-tolerance

Quantification of the performance of the superadiabatic geometric gates is obtained via standard quantum process tomography (QPT)32 measurements, which allow to reconstruct the full experimental quantum process matrix χ exp and therefore to determine the quantum gate fidelity F = Tr(χ exp χ 0 ),33 where χ 0 is the theoretically anticipated process matrix (for details on the experimental QPT procedure see Supplementary Information and ref. 13). Due to their dynamic nature and finite time duration the QPT pulses are susceptible to errors and we obtain the corrected quantum gate fidelity value \(\tilde F = F/F_{{\mathrm{ID}}}\) by normalization with the fidelity of the identity operation. We determine the experimental gate fidelities of the SAGQG to be \(\tilde F_{\mathrm{x}}^{{\mathrm{SAGQG}}} = 0.994_{ - 0.031}^{ + 0.026}\) and \(\tilde F_{\mathrm{z}}^{{\mathrm{SAGQG}}} = 0.995_{ - 0.024}^{ + 0.021}\) for Pauli-X and Pauli-Z operations, respectively. Additionally, the Hadamard gate is realized by a rotation of π/2 around the y-axis (R y (π/2)) and a subsequent rotation by π around the z-axis (R z (π)), resulting in an experimental fidelity of \(\tilde F_H^{SAGQG} = 0.992_{ - 0.029}^{ + 0.022}\). These values clearly exceed the necessary fidelity threshold on the order of 1 − 10−2 for the implementation of state-of-the-art error correction codes based on, e.g., surface codes.34,35 The SAGQG concept thus qualifies as a promising candidate for the implementation of scalable quantum computing.

Besides the fidelity of the individual, logical gates, we additionally assess the average error probability over the set of universal gates employing randomized benchmarking.36 Based on the application of randomly assembled sequences of a set of logical gates, randomized benchmarking allows for a good estimation of the error scaling given a long sequence of quantum gates, as relevant for viable applications in longer quantum algorithms. Figure 3a presents the average fidelity as a function of the number of computational gates l. For the SAGQG we obtain an average probability of error per gate of \(\varepsilon _g^{{\mathrm{SAGQG}}} = 0.0013(3)\), whereas an identical analysis for a set of dynamic quantum gates represented by π and π/2-pulses reveals an average probability of error of \(\varepsilon _g^{{\mathrm{dynamic}}} = 0.023(8)\), i.e., the geometric-phase based SAGQG performs one order of magnitude better than its dynamic-phase based standard gate (see Supplementary Information for details). Our results suggest that the SAGQG is significantly more resilient with respect to the type of noise and parameter imperfections present in our experimental system than the standard realization of dynamic phase-based quantum gates. Since the longest sequence duration (in total 99 gates) is much shorter than the longitudinal relaxation time (T seq ≈ 32 μs << T 1 ≈ 14 ms), decoherence effects are expected to be negligible and parametric noise is assumed to be the main source of error. Our experimental findings demonstrate the intrinsical robustness of non-adiabatic geometric phase-based quantum gates with respect to certain, experimentally very relevant types of parametric noise. These experimental findings of a non-adiabatic geometric quantum gate (Table 1) are in accordance with theoretical predictions of robustness in the distinct, but related adiabatic geometric gates.6,8,37,38,39 This joint robustness trait can be attributed to the fact that both adiabatic and non-adiabatic geometric phases and holonomies are global features, which are intrinsically robust with respect to locally occurring parameter imperfections and noise leaving the state-space area enclosed by the trajectory on the respective projective space invariant.

Fig. 3 Robustness analysis: a The randomized benchmarking analysis reveals the decay of the average fidelity in dependence of the number of computational gates l for a set of SAGQG (orange) and a set of dynamic quantum gates (blue). The average probability of error per gate are \(\varepsilon _g^{{\mathrm{SAGQG}}} = 0.0013(3)\) and \(\varepsilon _g^{{\mathrm{Dynamic}}} = 0.023(8)\), respectively. Error bars represent the standard error of the mean. b Minimal value of τ in dependence on the free parameter Ω 0 and Δ 0 for a system with maximal Rabi frequency Ω max = 7 MHz. c Measured quantum gate fidelity F as a function of τ for three free parameter combinations indicated in b by A, B and C. Solid lines are a guide to the eye. Vertical dashed lines represent the numerically calculated minimal τ value fulfilling Ω S (t, τ, Ω 0 , Δ 0 ) ≤ Ω max Full size image

In the following we examine the fidelity performance of the SAGQG with respect to variations in the gate evolution time. This is important for two reasons: (1) In order to most efficiently exploit the coherence time of the qubit, we need to investigate the theoretical velocity limits and experimental performance of the SAGQG and aim for fast quantum gate performance. (2) We experimentally examine the intrinsical robustness of the SAGQG with respect to experimental parameter imperfections. In particular we analyse the SAGQG performance outside its optimal parameter specifications. The latter is particularly relevant for the common experimental case where the Rabi frequency (for practical reasons) obeys a maximum bound \({\mathrm{max}}_t\left( {{\mathrm{\Omega }}_S(t,{\mathrm{\Omega }}_0,{\mathrm{\Delta }}_0)} \right) \le {\mathrm{\Omega }}_{max}\) (for parameter dependences see 'Methods' section). Given such a practical maximum bound Ω max for the experimentally achievable Rabi frequency, in Fig. 3b we show a contour plot of the numerically determined minimal τ-value, denoted τ min (Ω 0 , Δ 0 ), fulfilling the necessary criterion \({\mathrm{max}}_t\left( {{\mathrm{\Omega }}_S(t,{\mathrm{\Omega }}_0,{\mathrm{\Delta }}_0)} \right) \le {\mathrm{\Omega }}_{max}\). We like to stress again, the τ min (Ω 0 , Δ 0 ) limit is not given by theoretical constraints related to the state evolution (e.g., adiabaticity), but it is merely defined by the experimentally achievable Rabi strength Ω max . The smallest, experimentally feasible τ-value is equivalent to 1/(2Ω max ) corresponding to the length of a π-pulse t π , ultimately limiting the SAGQG length to t Gate ≥ 2/Ω max = 4t π . For our experimental conditions the minimal gate length t Gate = 4τ π corresponds to t Gate = 284 ns. If τ were chosen smaller than τ min this would require \({\mathrm{max}}_t\left( {{\mathrm{\Omega }}_S(t,{\mathrm{\Omega }}_0,{\mathrm{\Delta }}_0)} \right)\) to exceed Ω max which—given experimental limitations on Ω max —cannot be fulfilled by any experimental parameter set. Forcing τ < τ min experimentally leads to a marked mismatch between required and actual value of the driving field strength Ω S (t), i.e., an inconsistent, erroneous driving field parameter set.

Table 1 Experimentally obtained corrected quantum gate fidelities \(\tilde F_{}^{}\) and average gate error ε g of the single-qubit SAGQGs Full size table