Measuring qubit stability in a gate-based NISQ hardware processor

Some of the most problematic issues that limit the implementation of applications on noisy intermediate-scale quantum machines are the adverse impacts of both incoherent and coherent errors. We conducted an in-depth study of coherent errors on a quantum hardware platform using a transverse-field Ising model Hamiltonian as a sample user application. We report here on the results from these computations using several error mitigation protocols that profile these errors and provide an indication of the qubit stability. Through a detailed set of measurements, we identify inter-day and intra-day qubit calibration drift and the impacts of quantum circuit placement on groups of qubits in different physical locations on the processor. This paper also discusses how these measurements can provide a better understanding of these types of errors and how they may improve efforts to validate the accuracy of quantum computations.

Data are available from the corresponding author upon reasonable request.

PD was supported in part by the U.S. Department of Energy (DoE) under award DE-AC05-00OR22725. KYA and RCP were supported by the Quantum Information Science Enabled Discovery (QuantISED) for High Energy Physics program at ORNL under FWP number ERKAP61 and used resources of Oak Ridge Leadership Computing Facility located at ORNL, which is supported by the Office of Science of the Department of Energy under contract No. DE-AC05-00OR22725. YM and EG are supported by a Department of Energy QuantiSED grant DE-SC0019139. ZP gratefully acknowledges funding support from the NSF with an NSF pre-doctoral fellowship. We acknowledge the use of IBM Quantum services for this work, especially discussions with Nathan Earnest-Noble, Matthew Stypulkoski, Azia Ngoueya and Patrick Mensac from IBM. We thank North Carolina State University (NCSU) for access to the IBM Quantum Network quantum computing hardware platforms through the NCSU IBM Quantum Hub and thank IBM Research for the extended dedicated mode reservations and availability of the ibmq_boeblingen processor on which the computations were performed. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum team. The project team acknowledges the use of True-Q software from Keysight Technologies and useful discussions with Ian Hincks, Dar Dahlen and Arnaud Carignan-Dugas from Keysight.

1. Kübra Yeter-Aydeniz

   Present address: Emerging Engineering and Physical Sciences Department., The MITRE Corporation, 7515 Colshire Drive, McLean, VA, 22102-7539, USA

2. Erik Gustafson

   Present address: Theoretical Physics Division, Fermi National Accelerator Lab, Batavia, IL, 60510, USA

Authors and Affiliations

1. Physics Division and Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

   Kübra Yeter-Aydeniz

2. Department of Computer Science, North Carolina State University, Raleigh, NC, 27695, USA

   Zachary Parks & Aadithya Nair Thekkiniyedath

3. Department of Physics and Astronomy, The University of Iowa, Iowa City, IA, 52242, USA

   Erik Gustafson & Yannick Meurice

4. Department of Physics, North Carolina State University, Raleigh, NC, 27695, USA

   Alexander F. Kemper

5. Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA

   Raphael C. Pooser

6. Electrical and Computer Engineering and Department of Physics, North Carolina State University, Raleigh, NC, 27695, USA

   Patrick Dreher

Contributions

P.D. led the team, designed the operational implementation, coordinated with IBM Research for dedicated access to the IBM quantum hardware platforms and was a major contributing author for this paper. KYA ran the cycle benchmarking simulations and constructed the CB and QCAP graphs and made substantive contributions to the text. ZP captured the daily IBM backend property data. ZP also ran the TFIM simulations and wrote the analysis software for post-processing and plotting the TFIM data. EG provided the original TFIM code and the physics model circuits. ANT built the tables in the paper. AFK, RP, YM and PD edited and reviewed the document.

Corresponding author

Correspondence to Patrick Dreher.

Conflict of interests

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.

Appendix A: Daily ibmq_boeblingen qubit re-calibration schedule

For this specific project, IBM agreed to supply our team with approximately 140 h of dedicated reservation time and to follow an agreed upon customized calibration schedule. The customized schedule for dedicated time included a period in the morning from 4 am until 10 am and again in the afternoon from 3 pm until 11 pm. The complete ibmq_boeblingen re-calibration for both single and two-qubit gate gates was scheduled at 4:00 am ET, the beginning of the morning dedicated reservation time. A second re-calibration for only two-qubit gates ran at 6:00 pm ET, approximately 3 h into the afternoon dedicated reservation time. The calibration jobs took approximately an hour and a half to complete. Our team executed no external jobs on the device during the calibration process, allowing the calibration jobs to run without interference.

The single-qubit calibration process consisted of Ramsey and Rabi experiments to measure the frequency and amplitude of each qubit along with calibration of the optimal scaling factor of the Derivative Removal by Adiabatic Gate (DRAG) pulse used in single-qubit gates on superconducting hardware. The T1/T2 coherence times and measurement error rates of each qubit were also measured and recorded. Randomized benchmarking of the single-qubit gates was then performed in batches of non-adjacent qubits. The two-qubit calibration process was done in a similar manner. Calibration of the amplitude and phase of each pulse was completed before performing randomized benchmarking in batches of well-separated gates of similar length in order to measure the average gate fidelities. Each time the ibmq_boeblingen quantum computing hardware platform was re-calibrated and benchmarked, IBM published and made these backend properties available through Qiskit, the open-source quantum software development kit.

Appendix B: Cycle Benchmarking and Quantum Capacity (QCAP) protocols

This appendix summarizes both cycle benchmarking and quantum capacity protocols and their True-Q software implementation and parameter settings used for the computations reported in this paper.

1.1 Appendix B.1:  Cycle Benchmarking

Cycle benchmarking (CB) is a scalable noise characterization protocol that was selected to identify local and global errors across multi-qubit quantum processors. The CB protocol can measure errors such as process infidelity containing any combination of single gates, two-qubit gates and idle qubits, across an entire quantum device. CB helps keep track of each twirling gate and makes the process scalable with the number of qubits [77].

This protocol has the feature that the number of measurements required to estimate the process fidelity to a fixed precision is approximately independent of the number of qubits and is also insensitive to State Preparation and Measurement (SPAM) errors. Robustness to SPAM is very important characteristic because these type of errors can dominate the gate error measurement.

The CB protocol is presented in detail in reference  [20] and is schematically represented in Fig. 10. In CB, a gate cycle is an arbitrary set of native operations that act on a quantum register within a single clock cycle of time. Furthermore, within the CB protocol, there is a distinction between operations that can be physically implemented with relatively small and large amounts of noise, respectively, called ‘easy’ and ‘hard’ gate cycles.

The box on the left hand side of the figure shows the CB protocol “dressing” a primitive gate cycle of interest ( represented by \(\mathcal {\tilde{G}}\) ) by composing the cycle with independent, random n-qubit Pauli operators in such a way that the effective logical circuit remains unchanged. In the figure, the block \(\mathcal {\tilde{G}}\) represents the noisy implementation of the gate(s) being measured in the circuit. The blocks \(\mathcal {\tilde{R}}_{i,j}\) are random Paulis represented by the jth tensor factor of the ith gate inserted into the cycle to create an effective Pauli channel for the gate \(\mathcal {\tilde{G}}\) being measured. The blocks \(\mathcal {\tilde{B}}\) and \(\mathcal {\tilde{B}}^{\dagger }\) represent basis changing operations connected with controlling SPAM errors.

CB decouples state preparation and measurement errors from the process fidelity estimation of a particular gate cycle by applying the noisy, dressed cycle to the system m number of times, (called the sequence length) and extracting the process fidelity from the average decay rate as a function of this sequence length. This Pauli twirling of gate cycles map coherent errors into stochastic Pauli errors, which are then measured in the prepared eigenstates of the Pauli basis set.

This is represented by the top box in the center of the figure showing all measured Pauli decay expectation values plotted as a function of the gate sequence length. In practice, this process is computed using at least three distinct gate sequence lengths. Each measurement sequence produces an exponential decay of the expectation value versus the sequence length. Taken together, the set of exponential decays of the form \(Ap^{m}\) can be fit to the cycle of interest as a function of the circuit depth for each basis preparation state.

Using the fitted exponential decay, the individual process infidelity for each Pauli Decay term \(e_{F}\) can be measured as shown in the box on the right hand side of the figure. An average process infidelity and error for the particular cycle \(\mathcal {\tilde{G}}\) is calculated and is represented by the solid line and shaded band on the graph.

Block diagram of the CB protocol implementation

Full size image

For our project in order to measure the error characterization associated with the two-qubit gates in the TFIM circuits, the cycle benchmarking (CB) protocol was implemented using the True-Q software package. This package included a function make_cb that can produce quantitative measurements showing the effect of global and local error mechanisms affecting different primitive cycle operations of interest using CB.

The make_cb in True-Q uses a set of input parameters for the calculation. The first parameter is the cycle of interest. The second parameter sets how many times to apply the dressed cycle to observe the decay of the expectation values. Here, dressed cycle is the term that is used for denoting the target cycle preceded by a cycle of random elements of the twirling group. The number of random cycles need to be chosen carefully such that exponential decay is evident and the fidelity can be accurately estimated.

The third parameter in the function is the number of circuits for each circuit length determined in the second parameter, i.e., the number of random cycles. The last parameter in the function is the number randomly chosen Pauli decay strings. One can also specify the twirling group to be used that will be used in the process to automatically instantiate a twirl based on the labels in the given cycles. The supported twirling groups in True-Q software are Pauli, Clifford, unitary and identity. The software also offers initializing a twirl with single-qubit Cliffords. After the circuits are generated using this function, the expectation values of the Pauli operators are calculated which then gives the process infidelity for the cycle of interest by using an exponential fit to the decay of the expectation values.

The Clifford (C1) gates for the hard gate twirling were selected to minimize the computation time so that they would complete within the morning and night dedicated time windows available on ibmq_boeblingen. The C1 twirling used random single-qubit Cliffords which had the effect of symmeterizing the X, Y and Z noise. This ultimately allowed for an analysis of the depolarization error, which is one of the simplest of the systematic errors to measure and study.

To calculate the contribution of each of the Pauli decay terms to the average process infidelity \(e_F\), C1 twirling was done using gate sequence circuit lengths of 2, 10 and 22. Random Clifford gates were applied to each of the different pair combinations of CNOTs. Here, the sequence length refers to the number of times the cycle of interest appears apart from state inversion. We used 48 random circuits in each sequence length and 128 shots. The combination of the CNOT gate being measured and the sequence of random Cliffords defines a dressed cycle of the CNOT gates being measured. For each of the three different circuit lengths, the expectation values were calculated for all 16 of the Pauli decay terms. From these expectation values, fits to the exponential decay \(A p^m\) (SPAM parameter A and the decay parameter p) are calculated for each Pauli decay term.

Individual process infidelity measurements were recorded for every CNOT pair for each of the three different qubit layouts on ibmq_boeblingen device as shown in Fig. 2. For example, on Layout 1 measurements included all of the combination of two-qubit cycles ([0, 1 and 2, 3], [0, 1], [1, 2] and [2, 3]). Similar measurements were taken on the CNOT cycles for Layouts 2 and 3. Hence, there are four cycles studied for each qubit layout

The average process infidelity of the dressed cycle for that CNOT pair was computed based on the calculated values of each of the Pauli decay terms. Both the individual process infidelity and average process infidelity measurements were computed and used in the stability analysis of the qubits on the ibmq_boeblingen processor. The individual process infidelities for each CNOT pair and the overall process infidelity are shown in Fig. 4 for inter-day and Fig. 6 for intra-day computations.

1.2 Appendix B.2:  Quantum Capacity

The QCAP protocol was used for comparing the measured performance of a circuit that is loaded onto a quantum computing hardware processor to the measurement of an equivalent idealized version of that same circuit. A value of “0” for a QCAP result means that the circuit being tested is identical to its idealized equivalent, whereas a QCAP value of “1” implies that the circuit being measured has no equivalent performance characteristics to its idealized equivalent. An increase in the QCAP bound as a function of evolution time is a measure as to how many time evolution steps can be included in a result before the signal being measured is overcome by noise in the circuit.

For the actual QCAP measurements, the make_qcap and qcap_bound functions in the True-Q software were used to obtain an equivalent bound on the performance of a circuit as if it were computed using randomized compiling for calculating the process infidelity of the entire circuit of interest. The parameters to generate the collection of circuits for the Quantum Capacity bound make_qcap function use similar parameters as make_cb function, i.e., the circuit of interest, a list for the number of random cycles, number of circuits for each random cycle and total number of randomly chosen Pauli decay strings. After generation of quantum circuits, these circuits are embedded into qcap_bound as well as the circuit of interest to return a bound on the circuit performance. In this particular project, due to limited access to the dedicated mode on ibmq_boeblingen device we used sequence lengths of 4 and 16. The number of random circuits in this case is 30, and each of these circuits was run \(N_{\text {shots}}=128\).

We selected three separate groups of qubits on the ibmq_boeblingen hardware platform as shown in Fig. 2 to study the error characterization of TFIM Trotterization circuits using CB. The quantum circuit for evolution under the TFIM Hamiltonian has three pairs of two-qubit CNOT gates (c.f. Fig. 1). For the QCAP computation, we selected Circuit 1 shown in Fig. 1 in order to compare to previous TFIM measurements [54]. We computed an estimate to the QCAP bound of the circuit 1 CNOT cycles in the TFIM Trotterization quantum circuits as a function of the number of Trotter steps. We also calculated the QCAP bound from the CNOT error rates reported by IBM using RB. To this end, we used the expression for the relationship between the average process fidelity and the average gate fidelity as seen in Eq. (6). For a quantum circuit with N CNOT gates (Eq. B.1), the QCAP bound is calculated using CNOT error rates provided by IBM.

The QCAP bound versus step size was then plotted as the average process infidelity (QCAP bound) as a function of number of Trotter steps (as a function of time). From this graph, the performance of the circuit implemented on the set of specific qubits on that specific hardware platform can be measured over time.

Appendix C: Tables

See Tables 1, 2 and 3.

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions