Advancing quantum algorithms for large-scale simulations of quantum spin chainsQiskitQuantum Error Correction & MitigationQuantum ResearchQuantum Use Cases

Many trace the birth of quantum computing to 1981, when physicist Richard Feynman declared, "if you want to make a simulation of nature, you better make it quantum mechanical." More than 40 years later, quantum simulation continues to be one of the most promising applications for quantum computers. The difference is that now we have utility-scale quantum hardware with 100+ qubits and high enough quality that we can run quantum simulations that are impractical to address with classical methods.

Eventually, researchers hope quantum computers will simulate quantum systems that are entirely beyond classical methods. To get there, we'll need to continue improving not only quantum hardware, but also quantum algorithms, which are hindered by errors caused by noise. For quantum computers to reach their full potential, it's crucial that researchers use utility-scale quantum hardware to run large-scale simulation experiments and continue developing useful quantum algorithms. This means devising novel quantum algorithms, and it also means finding ways to combine and enhance established methods. In either case, the goal is to build algorithms that both take full advantage of the utility-scale devices we have today, and which are also capable of scaling to the more powerful quantum processors of the future.

In this blog post, we'll take a look at an experiment in which researchers from Stony Brook University in New York take an important step toward doing exactly that, demonstrating results from simulations of interesting quantum systems using variational quantum algorithms implemented across up to 102 qubits on IBM's superconducting quantum computers. Their research was published last year in Physical Review Research. While the Stony Brook team's experiments don't yet reach beyond classical brute force simulation, since they are based on relatively low-depth quantum circuits, this work still represents important progress toward large-scale implementations of the variational quantum algorithms that today are studied widely in quantum simulation research.

Below, we'll explain some of the key techniques that enabled the authors to obtain accurate ground state energies of quantum systems using quantum computers. In particular, we'll explore the researchers' use of physics-motivated ansatzes - more on those later - as well as the protocol they used to perform their energy measurement experiments, and their approach to error mitigation.

Physics-motivated ansatz for spin chain simulation

For this study, the authors focused their attention on XXZ and Heisenberg spin chains, two simplified models of magnetism

. You can envision a spin chain as a system of particles with "spin," - or in other words, particles with "intrinsic angular momentum" - all arranged together in a lattice, much like how particles are arranged together to form different kinds of materials.

By analyzing the way the spins in these spin chains behave and interact with one another, physicists can learn a great deal about the quantum phases of matter in quantum systems. Quantum phases of matter are similar to more commonly known states of matter like liquid or gas. Physicists study quantum phases of matter and the way they transition from one phase to another in much the same way a chemist might study how a liquid water transitions into water vapor.

The Stony Brook researchers set out to simulate large spin chain systems on quantum hardware, and to use these simulations to calculate the ground energy of those systems. Ground energy is the energy of a quantum system when it is not in an excited state and is instead in equilibrium, i.e. when the system is at its lowest possible energy.

The layout of the 127-qubit backend, with a chain of 102 qubits illustrated by the thick shaded line in red. Each qubit in the chain represents the spin of a particle in the spin chain simulation.

The authors began their experiments by implementing an ansatz motivated by the physics of the XXZ and Heisenberg models. In quantum computing, an ansatz is a parameterized quantum circuit that serves as the first circuit run in the variational quantum algorithms we use to simulate quantum systems. An ansatz is essentially an "educated guess" at the optimal circuit and circuit parameters for simulating a particular quantum system, and a "physics-motivated ansatz" is simply an ansatz circuit whose design is informed by known physics theories. Variational algorithms are iterative, so after running the initial ansatz circuit once, the algorithm will use the outputs of that first circuit run to optimize the circuit parameters before running the circuit again. This iterative cycle continues until the algorithm reaches an optimal output.

The Stony Brook researchers designed their ansatz by first starting with the Hamiltonian

Hamiltonian. The Hamiltonian of the model is the mathematical description of its total energy. For the XXZ model, it's the sum of the terms representing different interactions between adjacent spins. In quantum mechanics, each Hamiltonian corresponds to a distinct energy spectrum, that represents the set of possible total energies of the system. The quantum state with the lowest energy is the 'ground state,' while the others are 'excited states.' Understanding the ground state properties of a system is crucial because its behavior at sufficiently low temperatures can be inferred from the properties of its ground state.

H_xxz, which represents the total energy of the XXZ model. They then extracted the terms of the Hamiltonian corresponding to the interaction between odd-index spins in the spin chain (i.e. the first qubit in the spin chain, the third, etc.) and their adjacent spins. This new Hamiltonian, called H_odd, has a known ground state |Ψ_singlet⟩.

From there, the authors defined a new parameterized Hamiltonian H(s)=(1-s)H_odd + sH_xxz, which is a linear interpolation between H_odd and H_xxz. They reasoned that if you start with the ground state of H(s) for s=0 (i.e., |Ψ_singlet⟩), and are able to stay in the ground state while increasing s - a parameter representing time steps in the evolution of the quantum state - you will get the ground state of H_xxz at s=1, which is the result you want to achieve.

This means you can adopt an ansatz inspired by the previous intuition: The ansatz begins as an implementation of |Ψ_singlet⟩ using the appropriate quantum logic gates. That's followed by the evolution of the system according to the terms in the Hamiltonian H(s), which gradually increases the value of s up to s=1. If you manage to keep the system in the ground state during this evolution (provided there is an energy gap along the path), as it was in the beginning with s=0, you will get the ground state energy of H_xxz when s reaches 1.

To implement this idea in the form of a gate-based parameterized quantum circuit, the authors discretized the Hamiltonian using an approximation called Trotterization. Moreover, they turned this discretized evolution into a variational form allowing more flexibility.

This means that many of the quantum gates implemented in the quantum circuit depend on parameters, and these parameters can be tuned to minimize the measured energy. The possibility of modifying the angles of the parameterized quantum gates, with respect to the fixed angles used in the original discretization, may compensate for the approximation error.

(Fig.1a of the paper.) Scheme of the variational ansatz. The state is initialized as the ground state of H_odd. The remaining gates represent the variational form of the circuit (see main text).

Measurement protocol and error mitigation

Once they reached the desired quantum state, the authors used Bell-state measurements on nearest-neighbor qubits to evaluate its energy. Bell-state measurements are joint measurements of quantum states involving two qubits. Because qubits can only be measured in the 0 or 1 state, a Bell-state measurement implemented in a quantum computer only has four possible computational outcomes - |00⟩, |01⟩, |10⟩, or |11⟩.

For the XXZ model, each of these four Bell-states has a specific energy. The energy contribution of the bond between two nearest-neighbor qubits corresponds to the energy of the Bell-state obtained from the measurement outcome. The sum of all energy contribution corresponds to the total energy.

The authors divided all bonds into two groups and performed the Bell measurement in parallel within each group. In this way, they only needed to perform 2 sets of Bell measurements.

In a quantum chip, not all qubits are connected to each other. Instead, they are connected to their "nearest-neighbors." The white rectangles in this image highlight examples of nearest-neighbor qubits, which are any pair of connected, adjacent qubits. The authors performed two sets of measurement to account for all the possible pairs. Pairs shown in the figure can be seen as part of one of the two sets. The other set would include pairs 2/1, 0/14, etc.

Of course, the outputs of these measurements will include errors caused by noise. So, after measurement, the next step is error mitigation.

The authors began by applying a standard readout error mitigation

Readout error mitigation If you have a quantum circuit with few qubits, you can calculate the ideal output probability distribution P_ideal and then perform a reasonable amount of measurement in your real quantum hardware to obtain an experimental probability distribution P_measured. These quantities are related by the probability matrix M by the expression P_measured = M*P_ideal. By inverting this relation, you can use M to mitigate the noisy outcomes of your quantum device. However, for N qubits, the size of M is 2^N x 2^N, thus this approach is intractable for large number of qubits.

technique adapted to this case. Since the XXZ model contains only nearest-neighbor interactions, the authors focused on the mitigation of measurements on each pair of qubits in a bond. In this way, the probability Matrix M - which essentially represents the probabilities of the quantum algorithm outputting correct or incorrect results when run on quantum hardware - has dimensions 4×4 and can be calculated on each pair of qubits involved in the interacting Hamiltonian.

Representation of P_measured = M*P_ideal for 2 qubits. The matrix M here is 4×4 and each element Pr(a'b'|ab) is the probability of measuring state |a'b'⟩ with the noisy device instead of the true state |ab⟩. The diagonal highlighted in green is the probability that we measure the correct outcome in the bell measurement, while the rest highlighted in red is the probability that we measure an incorrect state.

This technique is particularly helpful when dealing with Bell-state measurements. That's because Bell-state measurements involve additional CNOT gates, which can be a source of noise. By using the technique described above, we can mitigate the noise caused by these imperfect CNOT gates, as well as the standard noise caused by the final measurement.

The calculated energy obtained from the variational circuit is affected by quantum gate errors. To mitigate these errors, the authors used the gate-level zero-noise extrapolation (ZNE)

Zero-noise extrapolation. The zero-noise extrapolation method is based on the following idea: if you apply an ideal quantum circuit U to the initial state |00…0⟩, you get |Ψ⟩ = U|00…0⟩, and this state is equal to |Ψ_n⟩ = U(U^-1 U)^n |00…0⟩ where n is a nonnegative integer. If you evaluate the expectation value of an observable O of the quantum state |Ψ⟩, ⟨Ψ| O | Ψ⟩, you should get the same result for different values of n. Unfortunately, the noise acting on the quantum gates influences the output of the quantum circuit. For this reason, by increasing n, you also increase the amount of noise. You can define the noise level m = 1+2n, where n=0 corresponds to the quantum circuit U applied to the initial state, so the noise level is equal to 1. Since you can calculate the energy for different values of n just by repeating the gates of your variational circuit U (and its inverse U_-1), you can obtain different values of energies for different levels of noise. At this point, you can use this data to extrapolate the expectation value at the zero-noise limit m=0.

method. However, they didn't stop there.

Example of zero noise extrapolation (ZNE). After getting the noisy results for m=1, m=1.2, and m=1.6, you can extrapolate to m=0, i.e. zero noise.

The authors were able to improve the ZNE results by using a reference state. The energy related to |Ψ_singlets⟩ is exactly known, so they tuned the parameters of the circuit to get the correct result, and then calculated the ratio between the known ideal energy value and the ZNE-mitigated energy value obtained from their experiments. Then, since this factor represents the "difference" between the ideal value and the mitigated one, they used it to rescale all the other mitigated results they obtained from their parameterized quantum circuit.

Results

Let's review everything the authors achieved with their experiment. They initially set out to calculate the ground energies for large-scale versions of the XXZ and Heisenberg quantum spin chain models. They found the optimal parameters for the quantum circuit they would use to simulate these quantum systems by following a variational process

Variational process. A variational process is an iterative process that consists of tuning the parameters to decrease the value of the measured energy. For each iteration of the variational process, the energy is measured and a new set of parameters with lower energy can be calculated. For the next steps, the process is repeated until a specific condition (e.g. a certain level of accuracy) is met.

, where each iteration of their circuit was simulated with a classical matrix product state (MPS) method.

This approach allowed the authors to evaluate the accuracy of their optimal ansatz with respect to the ideal ground state without taking into account the noise of the quantum computer. They ultimately obtained good accuracies, especially when increasing the number of gate layers, as shown in the figure below.

(Fig.3b of the paper.) The relative error between the ground state energies and the energies obtained with the classical simulation of the quantum circuits with optimal parameters. The errors are displayed as a function of the number of qubits N and as a function of the number of layers of gates.

From there, the authors implemented the quantum circuit with the optimal parameters on real IBM quantum computers, and they used readout error mitigation and ZNE error mitigation - the latter enhanced through the use of reference state - to obtain the desired results. Using 102 qubits for the computation, the researchers calculated a result that exhibited a difference of only 1.42% from the exact ground-state energy of the XXZ model.

Another way to understand the authors' workflow is by thinking about it in terms of Qiskit patterns, a four-step framework for solving problems with quantum computers:

1. They mapped their problem - finding the ground state energy of a quantum model - to a variational process involving parameterized quantum circuits with an ansatz motivated by the physics of the problem.

2. They optimized their circuit for the target quantum hardware by using the basis gates of the quantum computer and by minimizing the number of shots required using Bell measurements.

3. They executed their circuit on the quantum hardware.

4. They post-processed the results with the proposed readout error mitigation and zero-noise extrapolation, the latter of which they improved further by using a reference state.

The authors also note that it is entirely possible to perform the variational process from their experiment using only quantum simulation methods, and without the aid of classical techniques such as MPS. They made the case for this claim by showing the low energy error obtained for circuits with random parameters, which are similar to the circuits obtained at the beginning of a variational process.

Another notable result came from the author's work on the Heisenberg spin chain. The team successfully implemented their procedure for not only the XXZ model, but also for Heisenberg spin chains across several system sizes ranging from 4 to 102 qubits. They then took the approximate ground state energy per site - an estimate of the average ground state energy per particle - and extrapolated that number to a Heisenberg spin chain of infinite length. This theoretical, infinitely long spin chain is also known as an "infinite system limit."

Infinite system limit. The infinite system limit is a useful concept here because it is often possible to calculate exact solutions to quantum simulation problems when they concern quantum systems that approach infinite size in terms of volume and number of system particles, but which remain more or less constant in particle density.

In the Heisenberg case, the authors found that the values obtained from their extrapolations showed a difference of only 4% from the results obtained with the Bethe ansatz, a theoretical tool for finding the exact solutions for simulation problems involving quantum many-body models like the Heisenberg spin chain.

Findings like these serve as yet another compelling data point demonstrating the ability of utility-scale quantum hardware to deliver results that show strong agreement with established physics theory. Taken as a whole, the Stony Brook team's spin chain experiments bring us one step closer to using quantum computers to study the properties of useful quantum systems. This future capability will have enormous implications for both theoretical physics and practical applications such as materials science.

Using Qiskit to reproduce the ZNE with reference states method

To wrap things up, let's look at a simple implementation of the Stony Brook University research team's zero-noise extrapolation method with reference states using Python and Qiskit.

For the sake of simplicity, this example uses a toy model as a parameterized quantum circuit whose noisy outputs we can mitigate using the author's combination of the ZNE method and the reference state. Although it is not the same circuit implemented in the paper, the method shown is general and can be extended to other simulations.

Clever, custum implementations of quantum error mitigation

Error mitigation in Qiskit Runtime. We should note that you can also run utility-scale experiments using the pre-packaged quantum error mitigation methods available in the Qiskit Runtime Estimator, which offers built-in measurement error mitigation, ZNE, and other methods. Check out this tutorial to get started experimenting with Qiskit Runtime error mitigation methods.

, like the method developed by the Stony Brook University research team, will be essential as we begin exploring larger and more complex quantum systems with utility-scale quantum hardware.

Despite the enormous technical progress we've seen in quantum hardware over the past decade, we know their raw outputs are still heavily impacted by errors caused by environmental noise. To get results that more closely approximate the quantum systems we study, we need to continue investigating techniques that allow us to get as much as we can from quantum error mitigation and other post-processing techniques. This will continue to be true until we achieve quantum error correction, a future capability that could one day enable quantum computers to correct errors during the course of their computations.

The paper we've reviewed in this blog provides powerful examples of how to boost the effectiveness of these techniques by leveraging the known properties of the model, for example, by considering only the pairs of qubits for the readout error mitigation and using a known reference state to improve the results of zero noise extrapolation.

These kinds of physics-motivated insights, built upon decades of theoretical research, could play a crucial role in the future of quantum simulation. For more details on how this and the other techniques used in the Stony Brook University experiment might offer value for your own research, be sure to read the author's full paper.

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