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This post was contributed by Eyal Leviatan, Barak Katzir, Eyal Bairey, Omri Golan, and Netanel Lindner from Qedma, Joshua Goings from IonQ, and Daniela Becker from AWS.

Quantum computing is an exciting, fast-paced field. And especially in these early days, unfettered access to the right set of resources is critical in order to accelerate experimentation and innovation. Amazon Braket provides customers access to a choice of quantum hardware and the tooling they need to experiment, while also enabling them to engage directly with experts across the field from scientists to device manufacturers.

In this post, the team from Qedma, a quantum software company, dives into how they used Braket Direct to accomplish a milestone demonstration of their error mitigation software on IonQs Aria device. Leveraging dedicated access to quantum hardware capacity using reservations and collaborating with IonQ scientists for expert guidance directly via AWS, Qedma was able to successfully execute some of the most challenging Variational Quantum Eigensolver (VQE) circuits on a quantum processor to date.

In todays quantum processing units (QPUs), the susceptibility to various forms of noise results in errors that corrupt the quantum program and eventually render the results useless. The accumulation of errors over time, limits the duration and therefore the performance of quantum algorithms. Thus, achieving quantum advantage the ability to perform computations on quantum computers significantly faster than with classical supercomputers, needs a solution to mitigate the detrimental impact of these errors and enable algorithms to scale.

Error mitigation aims to reduce the effect of errors on the outputs of circuits executed on noisy quantum devices. However, these improvements come at the cost of runtime overhead that increases with the number of two-qubit gates (circuit volume) in the circuit. To overcome this, Qedmas novel approach to error mitigation, and the Qedma Error Suppression and Error Mitigation (QESEM) product, requires exponentially less overhead compared to other methods and suppresses errors at the hardware level to run longer programs while maintaining reasonable runtimes, potentially accelerating the path to quantum advantage.

Below we detail how QESEM was used in conjunction with IonQs Aria device via Braket Direct to produce high-accuracy results for a variety of quantum chemistry and quantum materials applications. We also show how Braket Direct provided us with dedicated QPU access, ideally suited for QESEMs interactive workflow, as well as the ability to connect directly with IonQs hardware experts. Scientific guidance from IonQ was important for tailoring QESEM to make the best use of Aria, and for constructing novel quantum chemistry circuits for the demonstration. These included VQE and Hamiltonian simulation circuits on 12 qubits, leveraging the high connectivity of IonQs devices. The results presented in this blog post demonstrate how users can push the boundaries of quantum chemistry and materials applications accessible on IonQs devices with Qedmas error mitigation, powered by Braket Direct.

QESEM can be used with any quantum program. When applied, QESEM first carries out a hardware-specific characterization protocol. According to the deduced error model, QESEM recompiles the input quantum circuit to a set of circuits that are sent to the device; the measurement outcomes are then classically post-processed, returning high-accuracy outputs, as we demonstrate below. The characterization process underlying QESEM ensures that its results are unbiased for any circuit. This means that QESEM provides results whose accuracy is only limited by the QPU time allocated for execution. In contrast, many error mitigation methods are algorithm-specific or heuristic. Algorithm-specific methods are not designed to mitigate generic errors across any quantum circuit, whereas heuristic methods generically converge to an incorrect (biased) output [1]. Relative to the leading unbiased and algorithm-agnostic methods, QESEMs QPU time is exponentially shorter as a function of circuit volume, as shown below.

We applied QESEM to three circuits from various applications and with a range of structural circuit properties (see Table 1). Specifically, we created a reservation via Braket Direct to get dedicated device access to IonQs Aria device. The reservation enabled the entire QESEM workflow to execute within a single working session where exclusive QPU access avoided the need to wait in line, and optimized throughput resulted in the shortest possible runtime. Along with the inherent stability of the physical properties of IonQs Aria, the reduced runtime ensured minimal drift of the system parameters during our experiments. This allowed QESEM to obtain an efficient description of the noise model during the execution.

Table 1: Properties of the circuits we demonstrated QESEM on.

Compared to the number of qubits they employ, all three circuits are comprised of a relatively high number of unique two-qubit gates between different pairs of qubits. This is made possible by the all-to-all qubit connectivity of IonQs hardware, which can calibrate an entangling gate between any pair of qubits; each of those gates is uniquely facilitated through the vibrational modes of the ion chain encoding the qubits. On the one hand, high qubit connectivity allows the compilation of complex circuits without incurring significant depth overhead. In contrast, on devices with lower connectivity, e.g., square lattice, applying a two-qubit gate to qubits that are not connected requires additional SWAP gates. On the other hand, the ability to run a large number of two-qubit gates poses a challenge for any characterization-based error mitigation method, since the noise model becomes very complicated. To address this challenge, QESEM used a characterization model specifically tailored to trapped ions, efficiently describing the errors of trapped-ion devices using a tractable noise model.

The first two circuits are examples of the VQE algorithm, which aims to find the ground state energy of a quantum many-body system, e.g., a molecule [1]. The specific examples we ran were designed to find the ground states of the NaH and O2 molecules. The third circuit realized a Hamiltonian simulation algorithm, implementing the time evolution of a quantum spin-lattice. We first describe the VQE circuits and focus on the oxygen molecule O2. Our efforts concentrated there due to its relevance to industrial and biological processes, while striking a balance between complexity and tractability making it a robust test for todays quantum devices. Moreover, the O2 experiment used a circuit volume of 99 two-qubit gates, larger than all VQE circuits featured in a recent experimental survey [3].

Typically, the presence of errors severely limits the size of VQE circuits because of the need for particularly accurate results. The ability to leverage the all-to-all connectivity of trapped-ion devices to reduce gate overhead is therefore well suited to this type of algorithm. With Braket Direct, we were able to incorporate expert guidance from IonQ on how to maximize the benefit of using their high connectivity and compile directly to their native gates to optimize the VQE circuits for the Aria device and produce the best results.

IonQ brought their quantum chemistry expertise to the table, equipping Qedma with circuits precisely crafted for the O2 molecule. Designed to mirror full configuration interaction results [4], these circuits included a chemistry-inspired Ansatz [5] supplemented by particle-conserving unitaries, which reflects the underlying molecular electronic structure. Additionally, IonQ undertook the classical optimization of the circuit parameters, setting the ground work for Qedma to apply QESEM effectively during the final energy assessment.

QESEM significantly enhanced the accuracy of the ground-state energy of the O2 molecule. Running this VQE circuit on Aria without error mitigation and measuring the ground state energy yields the result shown in red in Figure 1. This unmitigated result, i.e. executed without error mitigation, misses its mark by roughly 30%. In black, we show the exact energy, as it would have been obtained from the VQE circuit had it been run on a noise-free, i.e., ideal device. Using QESEM, the error mitigated energy (blue) closely matches the exact result up to the statistical error bar corresponding to the finite mitigation time. Moreover, the error bar accompanying the mitigated result is small enough to indicate a very clear statistical separation from the unmitigated result.

Figure 1: The ground state energy of the O2 molecule as obtained from running the VQE circuit on IonQ Aria without error mitigation (red) and with QESEM (blue) compared to the exact result that would be obtained on an ideal, i.e., noise-free, device.

Aside from the ground state energy, this VQE circuit also allows us to learn about the electronic structure of the O2 molecule. The states of individual qubits encode the electronic occupations of the molecules orbitals. A qubit in the 0 state signifies an empty orbital whereas the 1 state corresponds to occupation by a single electron. Moreover, from the correlations between pairs of qubits, we can extract the correlations between occupations. Some examples of occupations and their correlations can be seen in Figure 2. Again, all mitigated values match the ideal values up to the statistical error bars while the noisy results are, in most cases, far off.

Figure 2. Ideal, noisy and mitigated values for example orbitals occupations and their correlations.

Similar results for the NaH VQE circuit are shown in Figure 3. While the NaH circuit is narrower, i.e., involves fewer qubits, it requires a full qubit-connectivity graph and is of a comparable depth. Since this circuit only makes use of 6 qubits, the number of all possible outcomes is not very large, allowing the depiction of the full probability distribution of measurement outcomes (see Figure 3). Excellent agreement of the mitigated results with the ideal outcome can be seen for all bitstrings, demonstrating QESEMs capability to provide an unbiased estimate for any output observable of interest.

Figure 3: Results for the NaH VQE circuit. Left: The probability distribution of all possible measurement outcomes. Right: Observables of interest, e.g., the ground state energy. QESEM results (blue) reproduce the ideal values (black) up to statistical accuracy while the unmitigated results (red) are off.

In the study of quantum materials, there are two fundamental questions of interest: energetics and dynamics. The VQE algorithm presented above addresses the question of energetics. In contrast, the Hamiltonian simulation algorithm computes the time evolution of the quantum state of the material, i.e., its dynamics. The quantum circuit approximates the continuous dynamics by small discrete time evolution steps [6].

Spin Hamiltonians are widely used as models for quantum materials where the electrons are in fixed positions but interact magnetically. For this demonstration, we chose a canonical Hamiltonian, the so-called XY model with a perpendicular magnetic field [7]. The 12 spins, encoded by 12 qubits, reside on the sites of a three-by-four triangular lattice with periodic boundary conditions (see Figure 4). Under these conditions, the Hamiltonian simulation circuit requires high connectivity between the qubits to be compiled compactly. Beyond being a highly demanding benchmark, the Hamiltonian we simulated also illustrates rich quantum physical phenomena. The XY model is a model of strongly interacting bosons, as in a Josephson junction array. On a triangular lattice, this type of system can form an exotic phase of matter called a Supersolid [8].

Figure 4: Hamiltonian simulation. Left: the simulated triangular spin lattice. Colors represent different observables of interest the magnetization of individual spins (gray), and correlations between magnetizations of different spin patterns. Right: ideal, noisy and mitigated values for the different observables

Figure 4 shows the values of various observables of physical interest after one time-step (consisting of 72 two-qubit gates) is performed to an initial state where all spins, i.e., qubits, are oriented along the X direction. From left to right, these observables are the projections onto the X direction of the magnetization of single spins, and correlations of spin magnetizations along interaction bonds, lattice plaquettes, and strings of spins that envelop the lattice in one of its directions. Examples of each appear on the top panel in matching colors. These observables indicate the strength of various magnetic properties of the model. For each observable, we present the exact expectation values in black, the noisy unmitigated values in red, and the error mitigated results using QESEM in blue. Again, QESEM results reproduce the ideal values up to statistical accuracy, while the unmitigated results are statistically well-separated from both.

While we presented only a few specific examples, QESEM can be applied to any quantum circuit for which error-free results are desired. It is meticulously designed to optimize the accuracy-to-runtime tradeoff inherent to error mitigation methods. In particular, QESEMs QPU time, at a given statistical accuracy, scales exponentially better as a function of the volume of the target circuit compared to competing unbiased error mitigation protocols. For instance, a circuit with 120 two-qubit gates, run on a trapped-ion device with 99% two-qubit gate fidelity, would take 90 minutes to execute to 90% accuracy using QESEM, which can be easily completed within a two-hour device reservation using Braket Direct. The same circuit, executed with the leading competing unbiased and algorithm-generic error mitigation technique, Probabilistic Error Cancellation [9, 10], would take over a month.

Error mitigation is essential for executing cutting-edge applications on near-term quantum devices [1]. While the problems discussed in this blog can be simulated classically, QESEM enables accurate, error-free execution of large circuits increasing the number of two-qubit gates that can be utilized by more than an order of magnitude compared to unmitigated execution at the same level of accuracy.

Figure 5 shows the circuit volumes accessible with QESEM on trapped-ion devices. With expected near-future improvements in hardware fidelities and qubit counts, QESEM could enable executing generic quantum circuits faster than a supercomputer performing a state-vector simulation of the same circuit. Achieving this milestone will spur further exploration of applications requiring simulations of quantum systems, such as the design of novel materials.

Figure 5: accessible circuit volumes with QESEM on ion traps, assuming a desired accuracy of 90%. Active volume denotes the number of two-qubit gates within the circuit that affect the observable of interest. Here it is measured in terms of IonQs MlmerSrensen (MS) entangling gates. The black line estimates the time it would take a supercomputer to perform a state-vector simulation for a square circuit with the corresponding circuit volume. A square circuit consists of a sequence of layers in which each qubit participates in an MS gate, and the number of layers equals to the number of qubits (width=depth).

To learn more about Qedma and QESEM, visit Qedmas website. To further accelerate your research with dedicated access to quantum hardware including IonQs latest Forte QPU, check out the Braket Direct documentation or navigate to the AWS Management Console.

The content and opinions in this blog are those of the third-party authors and AWS is not responsible for the content or accuracy of this blog.

[1] Quantum Error Mitigation, https://arxiv.org/abs/2210.00921 (2022) [2] A variational eigenvalue solver on a photonic quantum processor, https://www.nature.com/articles/ncomms5213 (2014) [3] Orbital-optimized pair-correlated electron simulations on trapped-ion quantum computers https://www.nature.com/articles/s41534-023-00730-8 (2023) [4] Molecular Electronic-Structure Theory; John Wiley & Sons (2014) [5] Universal quantum circuits for quantum chemistry, https://doi.org/10.22331/q-2022-06-20-742 (2022) [6] Universal Quantum Simulators, https://www.science.org/doi/10.1126/science.273.5278.1073 (1996) [7] Boson localization and the superfluid-insulator transition, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.40.546 (1989) [8] Superfluids and supersolids on frustrated two-dimensional lattices, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.55.3104 (1997) [9] Probabilistic error cancellation with sparse PauliLindblad models on noisy quantum processors, https://www.nature.com/articles/s41567-023-02042-2 (2023) [10] Efficiently improving the performance of noisy quantum computers, https://arxiv.org/abs/2201.10672 (2022)

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