Emulating two qubits with a four-level transmon qudit for variational quantum algorithms

Using quantum systems with more than two levels, or qudits, can scale the computational space of quantum processors more efficiently than using qubits, which may offer an easier physical implementation for larger Hilbert spaces. However, individual qudits may exhibit larger noise, and algorithms designed for qubits require to be recompiled to qudit algorithms for execution. In this work, we implemented a two-qubit emulator using a 4-level superconducting transmon qudit for variational quantum algorithm applications and analyzed its noise model. The major source of error for the variational algorithm was readout misclassification error and amplitude damping. To improve the accuracy of the results, we applied error-mitigation techniques to reduce the effects of the misclassification and qudit decay event. The final predicted energy value is within the range of chemical accuracy.


I. INTRODUCTION
Quantum computing is widely considered a promising computing model due to its exponentially large Hilbert space [1].Most current quantum processors use two-level quantum systems, or qubits, as the basic building block for their ease of implementation, inheriting from the successful experience with binary classical computers.For a qubit processor made of N elementary building blocks, the size of the Hilbert space is 2 N .Current research focuses on scaling up the number of qubits N , but this approach faces challenges such as wiring [2], expensive control electronics [3], and frequency crowding [4,5].A complementary strategy is to increase the size of each building block by replacing the qubit with a qudit (dlevel system), which yields a Hilbert space of size d N .In current quantum processors, each building block requires a separate set of control electronics to operate, resulting in complex and costly systems that scale linearly with the number of building blocks.However, by using quditbased quantum processors, the number of required building blocks can be significantly reduced while maintaining the same size of Hilbert space.This also reduces the number of required control electronics, thereby simplifying the system and reducing financial costs.As a result, qudit-based approaches are a more practical solution for scaling up quantum processors in the near term.
The experimental use of qudits as the fundamental building blocks for quantum processors has been investigated in various systems, including photonic systems [6][7][8][9][10][11], ion traps [12,13] and nuclear magnetic resonance [14][15][16][17][18][19].Superconducting circuits, which are among the leading platforms for quantum computing, have also been explored for this purpose [20][21][22].The transmon is the most widely used building block in superconducting quantum processors [23], and it has more than two energy levels that can potentially be utilized for computation.However, these higher levels are more susceptible to charge noise and spontaneous decay, which presents a significant challenge [24][25][26][27].In recent years, transmons have been used as qutrits (three-level systems) in several studies, including the implementation of qutritbased quantum algorithms [28], simulation of topological Maxwell metal bands [29], the dynamics of quantum information in strongly interacting systems [30], and Tensor Monopoles [31].While these works used transmons with up to three levels of computational space, the results already demonstrate the potential of using transmons qudits as the fundamental building blocks for quantum processors.Previous work has also explored the use of higher levels of transmons beyond qutrits, and demonstrated quantum gates on 4-level transmons [32][33][34][35].Other applications on 4-level transmons have also been showcased, including enhancement of qubit readout fidelity [36] and encoding information for autonomous error correction [37].
The implementation of qudit-based quantum processors to execute quantum algorithms has two main challenges.The first challenge is that using qudits as building blocks requires a higher system quality, which is more difficult to fabricate and control [27,[38][39][40].For example, transmons have lower lifetimes at higher energy levels and exhibit larger charge dispersion.Implementing the readout of qudit typically presents more challenges due to the need to discriminate between more than two states from a single readout pulse.Therefore, a more careful design of the processor and signal processing is required to enable qudit readout.Another challenge for qudit-based quantum processors is that algorithms designed for qubit-based systems must be modified and adapted to fully exploit the increased Hilbert space provided by qudits.The algorithms must be compiled using a qudit gate set, which differs from the qubit gate set [41][42][43][44].Some well-known quantum algorithms, such as the Deutsch-Jozsa algorithm [45,46], Bernstein-Vazirani algorithm [47], Grover's algorithm [48], Quantum Fourier Transform [49,50], and Shor's algorithm [51], have direct generalizations of the qubit counterparts, which maintain the same principles but change the positional notation from the base-2 numeral system to the base-d numeral system.However, it is challenging to directly generalize an arbitrary quantum algorithm into a qudit version simply by changing the positional notation.Certain near-term applications are specifically designed to run on qubit-based quantum processors [52,53].For example, variational quantum algorithms for chemistry applications decompose the molecule Hamiltonian into a sum of tensor products of qubit Pauli operators.Generalizing these variational algorithms to work with qudits requires an innovative encoding mechanism.Upon submission, we noticed a parallel work demonstrates implementing variational algorithms that are modified for qudits [54].
The goal of this work is to address these challenges and provide solutions for the development of future quditbased quantum processors.To tackle the challenge of qudit readout, we demonstrate a high-coherence transmon device capable of utilizing its lowest four levels for information processing and enabling single-shot readout.For the challenge of developing qudit algorithms, we implement emulations of qubit systems using qudits.When the Hilbert space of N d qudit is equivalent to that of an N b -qubit system (2 N b = d N d ), the qudit can be used to emulate the N -qubit system.For example, a qudit with d = 4 can emulate a two-qubit system, as proposed in [32,33,55].By using this emulator, qubit algorithms can be executed directly on qudit-based processors without modification.These solutions bring us closer to developing practical and scalable qudit-based quantum processors.
In this paper, we present an implementation of a twoqubit emulator on a high-coherence transmon.Our transmon is designed to have the optimal operating parameters as a ququart (qudit with d = 4), and it is tuned to have single-shot readout capabilities that can distinguish the lowest four levels.To create the emulator, we compiled two-qubit operations into sequences of 4-level intrinsic operations that include only neighbouring transitions and virtual-Z gates.We characterized the performance of both the intrinsic and emulated gates using randomized benchmarking and gate-set tomography.We also implemented active resets on the qudit to prepare a high-fidelity initial state.We demonstrated the efficacy of a two-qubit variational quantum eigensolver algorithm on the emulator.We also analyzed the primary sources of error and implemented error mitigation techniques to obtain more accurate results.

II. IMPLEMENTATION OF THE EMULATOR
The two-qubit emulator is built on a single transmon with a coaxial geometry and off-chip wiring [56,57], where the lowest 4 quantum states of the transmon are used as the computational space.The same device has also been previously used as a qutrit [58].The two-qubit emulator maps the physical states |0⟩, |1⟩, |2⟩, |3⟩ of the transmon to the |00⟩, |01⟩, |10⟩, |11⟩ states of a virtual two-qubit device, see Figure 1(a).Single qubit gates for the first virtual qubit Q A are performed by driving transitions in the {|0⟩ , |1⟩} and {|2⟩ , |3⟩} subspaces in sequence.The single qubit operation on Q B can be implemented with a similar approach as for Q A by driving the two-photon transitions in the {|0⟩ , |2⟩} and {|1⟩ , |3⟩} subspace.However, it requires higher input power to drive the two-photon transition and tracking more parameters to implement the virtual Z gate.For simplicity, the gates are implemented by first performing an emulated SWAP gate, followed by applying the virtual qubit operations on Q A , and then apply another SWAP gate, as shown in Figure 1(b).The emulated SWAP gate is implemented by performing qubit-like X gate in the |1⟩ , |2⟩ subspace followed by two virtual Z gates to correct the phases.The iSWAP gate can be implemented with a similar scheme by performing qubit-like Y gate.Two-qubit gates are decomposed into single qubit gates and ZZ in- teractions, which are implemented using virtual Z gates.See the supplementary materials for further discussion of the qudit virtual Z gates.
The transmon is capacitively coupled to a resonator for dispersive readout.The readout pulse is a 10 µs square pulse with 1 µs hyperbolic-tangent rise and fall.4-level single-shot readout in this device is close to optimal due to the use of a readout resonator with linewidth κ/2π = 0.524 MHz and state-dependent frequency shift χ/2π = 0.288 MHz, where κ ≈ 2χ.Further details can be found in the supplementary materials.In the following experiments, we choose the single-shot readout frequency at 8782.41 MHz, see Figure 2(a).This choice provides a larger separation between the |2⟩ and |3⟩ states than between the |0⟩ and |1⟩ states, due to the larger standard deviation of the Gaussian distribution for the former pair of states, see Figure 2(b).This configuration is optimal for maximizing the signal-to-noise ratio.The frequency of the |0⟩ ↔ |1⟩, |1⟩ ↔ |2⟩, and |2⟩ ↔ |3⟩ transitions are 4134.33MHz, 3937.66MHz, and 3721.58MHz, respectively.For the intrinsic quantum gates, only the transitions between neighbouring states are used.
We measure energy relaxation times of T 2Echo = 76±27 µs, and T (23) 2Echo = 35 ± 14 µs.We observe a charge dispersion of 20 kHz on the |1⟩ ↔ |2⟩ transition, which is significantly lower than the Rabi rate of a single qudit pulse (which is 10 MHz for a 50 ns long π pulse).This implies that the charge noise contribution to the error is not detrimental to the implementation of quantum algorithms.See the supplementary materialsfor more details.

III. BENCHMARKING
The performance of the emulator is evaluated using gate-set tomography (GST) [59,60] and randomized benchmarking (RB) [61,62].GST is a method for characterizing quantum gates in detail.It can reconstruct the full Pauli transfer matrix (PTM) of the gate, as well as the initial state density operator and measurement operators that describe the state preparation and measurement (SPAM) errors.We utilize the pyGSTi tool [63] to implement GST on the two-qubit emulator.GST reconstructs the PTM and SPAM operators and shows that X 01 (π/2), X 12 (π/2), and X 23 (π/2) have infidelities of 4.976 ± 1.14 × 10 −3 , 2.966 ± 0.052 × 10 −3 , 2.906 ± 0.068 × 10 −3 , respectively.More details and discussions can be found in the supplementary materials.
After characterizing the initial state density operator, we observe a non-negligible population in the excited state, which prompts us to implement a qudit active reset protocol to prepare a high-fidelity initial state.We achieve this by utilizing the programmed FPGA to distinguish all four states using the nearest neighbor method within 40 ns, and sending a conditional sequence of π pulses in the {|2⟩ , |3⟩}, {|1⟩ , |2⟩} and {|0⟩ , |1⟩} subspaces to reset the state back to the ground state.The effectiveness of the active reset protocol is verified by characterizing the initial state operator using GST, see Figure 3.We find that the active reset significantly improves the initial state fidelity from 0.900 ± 0.011 to 0.9932 ± 0.0013, with twice repeated active reset being optimal.Prior to each following experiment, we apply the active reset protocol twice to ensure the preparation of a high-fidelity initial state.
To evaluate the performance of the emulator, we conduct RB on the virtual qubits A and B which allowes us to investigate the cost overhead of the emulation and determine its effectiveness.The result is shown in Figure   space are driven, a negligible population exists in this subspace.Therefore, we expect P to converge to 0.5, unlike the rest of the RB experiments that converges to 0.25.We also perform RB on both virtual qubits using the emulated single-qubit gates, yielding an average infidelity per Clifford of 3.14 ± 0.26 × 10 −2 .Finally, we implement full two-qubit Clifford randomized benchmarking.The two-qubit gates in this setup are randomly selected from the two-qubit Clifford group, which contains 11,520 two-qubit gates.Each Clifford gate is then decomposed into a sequence consisting of singlequbit gates acting on individual qubits, followed by a fixed two-qubit gate, and then another set of single-qubit gates acting on each qubit again.For this experiment, the fixed two-qubit gate is designed to execute the unitary operation U ZX = exp(−i(π/4)ZX), where ZX represents the tensor product of the Pauli Z and Pauli X operators.This operation is commonly realizable through cross-resonance interaction, which aligns with the conventional decomposition in fixed-coupling quantum processors.Then, these decomposed gates are mapped to the qudit intrinsic gate set.The single qubit gates are FIG. 5: Gate sequence for implementing the variational ansatz used to approximate the binding energy of a hydrogen molecule.R t are fiducial gates for implementing tomography.The gates are mapped to the qudit native operations under the rules described in Fig. 1, and the ZZ(θ) gate is implemented by applying two virtual Z gates Z 1 (−θ) and Z 2 (−θ).More details can be found in the supplementary materials .
mapped with the method described in Fig. 1, and the U ZX is implemented by driving a X gate on the {|2⟩ , |3⟩} subspace followed by a X(− π 2 ) gate on the first virtual qubit.We extract the average emulated two-qubit Clifford gate infidelity to be 9.51 ± 0.71 × 10 −2 .Our results suggest that emulating two qubits on a single qudit results in fidelity loss, indicating the need for optimal mappings of the operations on the emulated qubits to the qudit to further improve performance.

IV. VARIATIONAL QUANTUM EIGENSOLVER
The Variational quantum eigensolver (VQE) is considered a promising application of near-term quantum devices [52,53], which has been used to solve electronic structure problems such as the bonding energy of a hydrogen molecule.This particular problem requires a minimum of two qubits using the symmetry-conserving Bravyi-Kitaev encoding of the Hamiltonian [64][65][66].As an evaluation of the practicability of the emulator, we use a 4-level qudit system to emulate this two-qubit system and demonstrate VQE on a qudit.
To solve electronic structure problems, the VQE optimizes a wave function with a pre-assumed form, or "ansatz" |Ψ⟩.The variational principle guarantees that the ground-state energy of the electronic system denoted as E 0 , can be approximated by optimizing the parameters ⃗ θ in the ansatz |Ψ( ⃗ θ)⟩.It can be written as ⟨Ψ( ⃗ θ)| H |Ψ( ⃗ θ)⟩ ≥ E 0 , where |Ψ( ⃗ θ)⟩ is the ansatz parametrized by ⃗ θ, and H is the Hamiltonian.The wave function is typically generated using a parametrized quantum circuit, and in this case, we use the ansatz |Ψ( ⃗ θ)⟩ = e −iθXY |Ψ HF ⟩, which is suggested from the unitary coupled cluster theory [64,67,68].|Ψ HF = |01⟩⟩ is the Hartree-Fock state which is used as a starting point in this theory, and where it is assumed electrons occupy the lowest lying orbitals.
We implement the chosen ansatz on the two-qubit emulator using the gate sequence shown in Figure 5.
The energy of the hydrogen molecule is evaluated as , where P i ∈ {IZ, ZI, ZZ, XX, Y Y } and g i are coefficients that can be computed from the hydrogen bond distance R. The measurement of the quantum state gives the probability distribution of the state in |0⟩, |1⟩, |2⟩, and |3⟩, which we represent as a vector R = {p |0⟩ , p |1⟩ , p |2⟩ , p |3⟩ }.The expectation value for an operator M ∈ IZ, ZI, ZZ is evaluated as ⟨M ⟩ = diag(M ) T R. The ⟨XX⟩ and ⟨Y Y ⟩ are measured by swapping the X or the Y basis with Z basis of both qubits, respectively, and measure ⟨ZZ⟩.There is only one parameter in the ansatz; in the experiment, the parameter is swept using 100 points from −π to π.Each point is sampled 100 times, each time uses 500 shots to estimate the expectation values.The distribution of the estimated expectation value is displayed in the form of a heatmap in Figure 6.The solved energy versus the hydrogen bond distance is given in Figure 8.
In comparison to the simulation results, the experimental data displays some noticeable differences.As shown in Figure 6, we observe that the amplitude of the expectation value for all operators is lower than 1, whereas in the simulation it should be 1.Additionally, the ZZ expectation value exhibits a larger value when the parameters are set to 0 compared to π, which is not the case in the simulation where it remains constant.Moreover, the distribution plot for each expectation value shows a "shadow," indicating that some of the measured distributions are not well-approximated by a simple Gaussian distribution.See the supplementary materials for more details.These results suggest that the system is subject to a complex noise model.We consider two potential sources of error for this result: (1) amplitude damping error during the long measurement time, and (2) misassignment between the |2⟩ and |3⟩ states, as indicated by GST (see Figure 3(d)).We use simulations to investigate the impact of these two types of noise and compare the simulated expectation value with the experiment result.
To address the misclassification event between the |2⟩ and |3⟩ states, we introduce a readout model P (|2⟩) = (1−ϵ)P (|2⟩)+ϵP (|3⟩) and P (|3⟩) = (1−ϵ)P (|3⟩)+ϵP (|2⟩), where P (|i⟩) is the measured probability for state |i⟩ under the noise model.The blue lines in the simulation plots of Figure 6 represent the simulated expectation value versus parameter under the misclassification noise, with ϵ values of 0.1, 0.25, and 0.5, where lighter colours indicate lower noise strength and darker colours indicate higher noise strength.The simulation results reveal that this misclassification event is likely the cause of the observed "shadow" in the distribution of the expectation values.
For the amplitude damping channel, we introduce a completely positive trace preserving (CPTP) channel operator D for a qudit to describe the amplitude damping error [1,69].
Here we use √ γ ij |i⟩ ⟨j| , ∀i, j s.t.0 ≤ i < j < d − 1 and K 0 = |0⟩ ⟨0| + 1≤j<d−1 1 − ξ j |j⟩ ⟨j|.γ ji has real value, describs the decay rate from the j-th to the i-th level (ξ j = 0≤i<j<d−1 γ ji ≤ 1).We performed simulations using the error model described above and set γ ji = 1 − exp(−tΓ ji ), where 1/Γ ji is the effective T 1 for the {|i⟩ , |j⟩} subspace and t is the simulated waiting time for the decay of the system.The simulation results indicate that the distortion of the ZZ expectation value is similar to the simulated value from the amplitude-damping channel.This suggests that the amplitude damping error during the measurement is likely to be the main source of error for the ZZ expectation value.
To mitigate the effect of the misclassification error, a Gaussian fit is applied to the measured probability distribution to identify and remove the outliers that contribute to the "shadow" in the distribution [70].We exclude 50% of the population as outliers to visually eliminate all the "shadow", and show the remaining population in column (c) of Figure 6 ("Outlier removed").The resulting expectation values, shown in blue in Fig. 8, have significantly reduced error bars compared to the raw results.However, there is still a noticeable difference between the VQE re-sult and the Hamiltonian diagonalization result.
To mitigate both the readout assignment error and the amplitude damping error during measurement, we applied measurement assignment mitigation using the assignment matrix A. This matrix describes the probability of measuring the state |j⟩ when the system is prepared in state |i⟩, and is used to obtain a mitigated probability distribution R by inverting the assignment matrix as R = A −1 R. The mitigated expectation value was then evaluated as ⟨ M ⟩ = diag(M ) ⊤ A −1 R. The measured assignment matrix is shown in Fig. 7(a).Notable misclassification between the |2⟩ and |3⟩ states is observed, consistent with data from gate-set tomography.For comparison, we present the simulated assignment matrix, which accounts only for the T 1 decay event, as shown in Fig. 7(b).This matrix is evaluated as Ã = (I−Γ ⊤ ) t , where t is set to 10 microseconds (the measurement pulse length) to show a worst-case scenario.The error-mitigated expectation values are presented in Fig. 6 (c).We observed that the mitigated expectation value of ZZ and ZI exceeds the physical bond, and we suspect this is due to the difference between the GMM (Gaussian Mixture Model) classifier used for evaluating the expectation value and the GMM classifier used to evaluate the assignment matrix, see supplementary materials for more details.The latter classifier is trained on a different dataset collected after the raw expectation values are evaluated.We employ the mitigated expectation value without a hard physical bond to evaluate the energy, and the result is shown in green in Fig. 8.By applying the measurement assignment mitigation technique, we are able to decrease the difference between the averaged solved energy and the Hamiltonian Diagonalization result to reach the chemical accuracy threshold of 1.5 × 10 −2 Hartree [64].However, due to the large misclassification error, the error bar remains significant.It is worth noting that the assignment matrix already captures both misclassification errors and the amplitude damping error, the result would be overcorrected if we apply both assignment mitigation and outlier removal.

V. DISCUSSION
This work aims to explore the potential of higher levels in a transmon as a computational space for near-term quantum applications.By introducing higher levels, we increase the Hilbert space available for quantum algorithms, which can lead to more efficient and powerful algorithms.In this work, we demonstrate a high-coherence transmon device as a 2-qubit emulator to address these challenges.
We conduct detailed characterizations and benchmarking on both the hardware itself and the emulator, including active reset to prepare a high-fidelity initial state.We then implement variational quantum algorithms on the qudit device, confirming that charge noise at higher levels is not detrimental to the algorithm.The main source of error is the decay that occurs during the measurement process.We then mitigate this error with post-processing techniques and obtain a solution for the energy of a Hydrogen molecule with an error within chemical accuracy.
This study has focused on developing an emulator on a superconducting qudit.A future research direction is to implement entangling gates between ququart transmons.This direction could build upon earlier research that developed qutrit entangling gates [71,72].The next step would be to determine the most effective compilation method for these emulators, and demonstrate its universality.Additionally, further exploration of the impact of higher levels on the noise and error rates of the device could help improve its overall performance.Appendix A: Intrinsic SU(4) operations and virtual Z gates Physical gates Physical drives at the transition frequency between the |i⟩ and the |j⟩ states implement the X or Y rotation in a qubit-like subspace of the qudit.The parametrized SU(4) rotation operator for X-like and Y-like rotations are , respectively.The generators are shown as follows: The pulses employ a 50 nanoseconds Blackman-DRAG envelope, and they are separated by a 10 nanoseconds buffer time.
The Z rotation can be implemented virtually by shifting the phase of all gates in the rest of the sequence [75].In this work we define the Z gate notation as follows: The implementation of the Z rotation for a multi-level system is with an extension of qubit virtual Z gate.Suppose we would like to implement a Z gate in the following gate sequence.
where G k denotes a gate.Now insert an identity gate sequence Z m (θ)Z −1 m (θ) between all the following gates, and we get which is equivalent to the original gate sequence.Now we rewrite the G where G ′ k is implemented by shifting the phase of driving pulses.The last gate Z −1 m (θ) will not make any difference if the state is measured with an operator commutes with Z −1 m (θ), which is always the case for the standard dispersive readout.For a four-level qudit system, the virtual Z gate can be implemented as shown in table I .T (12)   2 Echo T (23)   2 Echo = 87±23 µs.To characterize the dephasing dynamics, spin echo experiments are performed on each neighbouring subspace to determine T 2 , which describes the phase coherence of the subspace.The normalized survival population P (|i⟩)N is defined as P (|i⟩) N = P (i)/(P (i) + P (i − 1)) to remove the effect of energy relaxation, where P (|i⟩) is the population ratio measured in state |i⟩ with i > 0. P (|i⟩) N is fitted to P (|i⟩) N (t) = e −t/T (i) 2 P (|i⟩) N (0), where T The charge-noise-induced error on the higher levels can be a problem for executing quantum algorithms.To measure the sensitivity of the charge noise, we implement a Ramsey interferometry experiment on the |2⟩ , |3⟩ subspace, as it is the most sensitive subspace among all three neighbouring subspaces of the four lowest transmon levels [77,78].Our results show that the frequency shift due to charge noise is around 20 kHz, which is significantly lower than the rabi rate of a single qudit pulse (which is 10 MHz for a 50 ns long π pulse).This implies that the charge noise contribution to the error would not be detrimental to the implementation of quantum algorithms.noise, which is independent in both the I and Q channels and has the same variance.Therefore for this work, the covariance matrices Σ k are selected to be diagonal and all the diagonal element are the same, ensuring the model to fit distributions that appear 'spherical' in the IQ plane.The implementation of the Gaussian mixture model is provided by the scikit-learn library [79].
Appendix H: Experiment setup

FIG. 1 :
FIG. 1: (a) The lowest four levels of the transmon are used as a qudit for quantum information processing.(b) The two-qubit operations are mapped to sequences of single qudit operations.R(θ) denotes an arbitrary single-qubit rotation operation, and R ij denotes the equivalent single-qubit operation in the {|i⟩ , |j⟩} subspace.X ij and Y ij and Z i are analogs of the single qubit X, Y, Z rotation in the SU (4) space, please refer to Appendix A for more details.

FIG. 2 :
FIG. 2: (a) Phase response of the readout resonator while the transmon is excited to the |0⟩, |1⟩, |2⟩, and |3⟩ states, respectively.(b) IQ plane of the single-shot readout.The states are prepared in the |0⟩, |1⟩, |2⟩, and |3⟩ states, respectively, denoted by the color.The four different levels were classified by a spherical Gaussian mixture model, where the cross denotes the centre of the Gaussian distribution, and the radius of the circle denotes three times the standard deviation.See appendix G for more details.
FIG.4: (a-b) Results of single-qubit randomized benchmarking on the virtual qubits A and B of the emulator, yielding average infidelity per single-qubit Clifford gate 1.91 ± 0.23 × 10 −2 and 2.89 ± 0.31 × 10 −2 , respectively.(c) Simultaneous randomized benchmarking of single-qubit gates on both virtual qubits, with gate sequences executed in series.The average infidelity per single-qubit Clifford is extracted to be 3.14 ± 0.26 × 10 −2 (d) Results of full 2Q randomized benchmarking on the emulator, and the average infidelity per two-qubit Clifford gate is 9.51 ± 0.71 × 10 −2 .

FIG. 6 :
FIG. 6: Experimental results are shown for the expectation values of each Pauli operator as a function of the ansatz parameter in the VQE used to determine the binding energy of a hydrogen molecule.The VQE ansatz is implemented using the gate sequence shown in Fig. 5.The white dashed curve represents the theoretically expected ground truth obtained by simulating the ansatz classically.The red curve represents the average expectation value of the distribution.Column (a) shows the raw experimental data.In column (b), the experimental data has been processed to remove 50% of the population using Gaussian distribution outlier detection to eliminate misclassification events.Column (c) shows the expectation values after applying assignment mitigation.Column (d) displays the simulated expectation values versus the parameter under the amplitude damping noise (red) and misclassification noise (blue).The colours from light to dark denote the strength of the error.The black line denotes the simulated expectation value without noise.

FIG. 7 :FIG. 8 :
FIG. 7: The assignment matrix measured during the execution of VQE sequences.(a) The assignment matrix was measured from the experiment.(b) the simulated assignment matrix that only accounts for the T1 decay event.The measured assignment matrix was used to implement measurement mitigation.

{|0⟩,
FIG. 9: (a) plot shows the spectroscopy phase response of a resonator coupled to a qubit.The resonator oscillation frequency shifts depends on the transmon's states, resulting in different phase responses shown in different colors.To read all four levels simultaneously, a readout signal is sent to the resonator with a frequency that maximally distinguishes the phase response of all four different states.(b) The scatter plot shows the I channel signal and Q channel signal, often referred to as the "IQ plane."The integrated sum of the demodulated signal over time gives a point on the IQ plane.The four different states correspond to four different regions on the IQ plane, whose size is determined by the measurement noise.The separation angle θ ij between region i and j is related to the frequency shift shown in (a).

FIG. 10 : 1 = 1 The T 1
FIG. 10: (a) to (c) show the population in each state as a function of the waiting time, when the transmon is initially prepared in the |1⟩, |2⟩, |3⟩ state, respectively.(d) to (f) shows a typical result Spin Echo experiment performed on the {|0⟩ , |1⟩}, {|1⟩ , |2⟩}, {|2⟩ , |3⟩} subspaces, respectively.The black marker in these figures represents the normalized success rate P = P (|i + 1⟩)/(P (|i⟩) + P (|i + 1⟩)), which is used to reduce the impact of the decay of the transmon during the experiment.(g) and (h) show the distribution of the effective T 1 and T 2 echo times, respectively, with repeated measurements.These experiments are repeated for 100 times.(i) displays the Fourier spectrum of the trace of the Ramsey interferometry experiment performed on the {|2⟩ , |3⟩} subspace over time.The frequency axis shows the measured frequency detuning from the transition drive frequency, while the experiment time axis denotes the time interval between the start of collecting the Ramsey trace and the start of the experiment.

FIG. 13 :
FIG.13:The ansatz presented in the native gate set.The first row in our implementation denotes all the physical gates that have been used, where R t represents the fiducials used for preparing the measurement basis.The dashed lines in the figure denote virtual Z gates that have been inserted into the circuit.The virtual Z gate sequences are described in the second row.

FIG. 14
FIG. 14: (a) and (c) show heat maps of the distribution of the expectation values of the IZ and ZZ operators, respectively, plotted against the ansatz parameter θ.Figures (b) and (d) display histograms of the distribution of expectation values along the white dashed line shown in figures (a) and (c).
4, showing that emulation causes fidelity loss.We generate single-qubit randomized Clifford gate sequences and convert them to the native qudit gate using the simulator's mapping rule.The RB sequences are run on the emulator, and the success probability P is defined as the population landing on the emulated state |00⟩.We fit the RB model to P = Ap m + B, where A, B, and p are fitting parameters, and m is the number of Clifford gates.We calculate the average infidelity per Clifford as r = (1 − p)(d − 1)/d, where d = 2 for single-qubit RB and d = 4 for two-qubit RB.The resulting average singlequbit Clifford gate infidelity for virtual qubits A and B are 1.91 ± 0.23 × 10 −2 and 2.89 ± 0.31 × 10 −2 , respectively.Notably, performing single-qubit RB on virtual qubit A only drives the population within the |0⟩ , |1⟩ subspace.Although transitions within the {|2⟩ , |3⟩} sub- Error generatorII IX IY IZ XI XX XY XZ YI YX YY YZ ZI ZX ZY ZZError GeneratorII IX IY IZ XI XX XY XZ YI YX YY YZ ZI ZX ZY ZZ FIG.12: Averaged gate infidelities, reconstructed process matrix and error generators of the gate set.The result is generated by pyGSTi maximum likelihood estimation.