http://dx.doi.org/10.1103/PhysRevA.110.012404

This research addresses the challenge of error mitigation in variational quantum algorithms (VQAs), particularly the Variational Quantum Eigensolver (VQE), which are essential for tackling problems in quantum chemistry, materials science, and optimization using NISQ computers. While VQAs offer some robustness against systematic errors, stochastic errors and limited coherence times remain significant obstacles.

### Unexpected Findings

**Linear Relationship:**The research reveals a surprising linear relationship between the energy estimated by a noisy VQE circuit and the total sum of gate errors in the circuit. This finding provides a simple and intuitive way to understand the impact of noise on VQA performance. For a six-qubit system, the error was reduced by two orders of magnitude using this linear extrapolation, decreasing from $10^{-3}$ to $3.7 \times 10^{-4}$ in units of transverse field strength.**Exact Extrapolation for Specific Circuits:**The researchers theoretically demonstrate that for certain types of variational circuits, specifically those with a structure corresponding to regular graphs (e.g., cycles), the proposed zero-noise extrapolation (ZNE) method can recover the exact noise-free energy in the limit of small errors. This suggests that the effectiveness of ZNE might be dependent on the specific circuit structure.**Robustness to Higher Noise Levels:**The numerical simulations show that the ZNE method remains effective even for relatively high noise levels, exceeding the range where the linear approximation is strictly valid. This robustness enhances the practicality of the method for near-term quantum computers. Specifically, for a 12-qubit system, the ZNE error remained smaller than the spectral gap even when the noise strength parameter ($max{q_{jk}})|T|d$) reached a value of 1, suggesting that the method can tolerate a significant amount of noise. However, the researchers observed that the standard deviation of the extrapolation error increased gradually with the number of qubits.

### Key Terms

**Variational Quantum Algorithm (VQA):**A type of quantum algorithm that uses a classical computer to optimize the parameters of a quantum circuit to solve a problem. Think of it as a hybrid approach that combines the strengths of both classical and quantum computing.**Variational Quantum Eigensolver (VQE):**A specific type of VQA used to find the ground state energy of a quantum system. This is important for understanding the properties of molecules, materials, and other quantum systems.**Zero-Noise Extrapolation (ZNE):**An error mitigation technique that runs an algorithm at different noise levels and then extrapolates the results to the zero-noise limit, aiming to approximate the ideal, noise-free output. Imagine drawing a line through noisy data points and extending it to where the noise should be zero.**NISQ (Noisy Intermediate-Scale Quantum) Computer:**The current generation of quantum computers with limited qubits and high error rates.**Qubit:**The basic unit of quantum information, analogous to a bit in a classical computer.**Quantum Gate:**A logical operation that acts on qubits, similar to logic gates in classical computers.**Gate Error Rate:**The probability of an error occurring during the execution of a quantum gate.**Hardware Efficient Ansatz (HEA):**A common type of quantum circuit structure used in VQAs, designed to be efficiently implemented on near-term hardware.**Depolarizing Noise:**A type of quantum noise that randomly flips the state of a qubit, like a coin flip deciding whether to change a 0 to a 1 or vice versa.**Spectral Gap:**The energy difference between the ground state (lowest energy state) and the first excited state of a quantum system. A larger spectral gap generally makes the system more robust to noise.**Circuit Error Sum (CES):**The total sum of gate error rates for all two-qubit gates in a quantum circuit, taking into account the specific mapping of abstract qubits to physical qubits.**Decoherence:**The loss of quantum information due to interactions with the environment. Imagine a qubit's delicate quantum state being disrupted by stray electromagnetic fields or temperature fluctuations.**SPAM Errors:**Errors that occur during state preparation and measurement (SPAM) - the processes of initializing qubits and reading out their final states. Think of these as errors in setting up the experiment and recording the results.**Crosstalk:**Unwanted interactions between qubits, where the operation on one qubit affects the state of another qubit. Like two radios interfering with each other's signals.**Ising Model:**A simplified model used in physics to study magnetism. It involves a chain or lattice of spins (like little magnets) that can interact with each other.**Hamiltonian:**A mathematical object that describes the energy of a quantum system. It's like a recipe that tells you how the different parts of the system interact and contribute to its overall energy.

### Approach

**Noise Model:**The researchers consider a noise model where two-qubit gates are subject to errors, while single-qubit gates are assumed to be noise-free. They assume that the error rate for a gate depends only on the specific pair of qubits it acts upon, reflecting the inhomogeneous nature of gate errors in real quantum devices.**Qubit Mapping and Noise Scaling:**They exploit this inhomogeneity to control the overall noise level in the circuit by varying the mapping between abstract qubits in the circuit and physical qubits on the quantum computer. Different mappings lead to different combinations of gate errors, effectively scaling the total noise in the circuit.**Linear Extrapolation:**They run the VQE algorithm for multiple qubit mappings, generating data points representing the estimated energy at different noise levels (circuit error sums). They then perform a linear fit through these data points and extrapolate the trend to the zero noise limit to approximate the noise-free energy.

### Results and Evaluation

**Linear Relationship Validated:**Numerical simulations confirm the linear relationship between the VQE energy and the circuit error sum for a range of problem instances and circuit depths, supporting the analytical predictions. For a six-qubit system, the error was reduced by two orders of magnitude using this linear extrapolation, decreasing from 10−3 to 3.7 × 10−4 in units of transverse field strength.**ZNE Accuracy:**ZNE significantly reduces the noise-induced error in the estimated energy. For a six-qubit Ising Hamiltonian, the extrapolation error was two orders of magnitude smaller than the minimum error observed in the noisy circuits.**Performance with Limited Permutations:**ZNE remains effective even when only a small fraction of all possible qubit permutations are used, making it more practical for quantum computers with limited qubit connectivity. For a 6-qubit system using an HEA with ring topology, the extrapolated energy converged towards the true ground state energy even with only a small set of permutations, exhibiting a standard deviation significantly smaller than the spectral gap of the problem.**Robustness to Higher Noise Levels:**The numerical simulations show that the ZNE method remains effective even for relatively high noise levels, exceeding the range where the linear approximation is strictly valid. This robustness enhances the practicality of the method for near-term quantum computers. Specifically, for a 12-qubit system, the ZNE error remained smaller than the spectral gap even when the noise strength parameter $(max{q_{jk}})|T|d$ reached a value of 1, suggesting that the method can tolerate a significant amount of noise. However, the researchers observed that the standard deviation of the extrapolation error increased gradually with the number of qubits.

### Practical Deployment and Usability

**Experimental Feasibility:**The proposed method for scaling noise by varying qubit mappings is experimentally feasible on current NISQ devices, as it only requires changing the assignment of abstract qubits to physical qubits.**Simplified Error Mitigation:**The linear relationship between energy and noise simplifies the ZNE procedure, making it easier to implement and potentially reducing the computational overhead compared to more complex extrapolation techniques.

### Limitations, Assumptions, and Caveats

**Simplified Noise Model:**The noise model considers only depolarizing errors on two-qubit gates, neglecting other noise sources, such as decoherence, SPAM errors, and crosstalk. Real-world noise is more complex, potentially limiting the accuracy of the extrapolation.**Scalability:**While the study demonstrates promising results for moderate-sized systems, the scalability of the method to much larger quantum computers with hundreds or thousands of qubits requires further investigation. The increasing homogeneity of error rates in larger systems could potentially degrade the effectiveness of the qubit mapping approach to noise scaling.**Amplified Errors in Specific Regions:**Biamonte and collaborators have explored the phenomenon of "parameter concentration" in QAOA [13], where optimal parameters tend to cluster within narrow ranges. This concentration could amplify the impact of noise, as small errors in these sensitive parameters could significantly degrade performance, potentially limiting the effectiveness of Uvarov et al.'s ZNE method.

### Correlation with Prior Research

**Akshay et al (2020, 2021) and Biamonte (2021):**Akshay and Biamonte's research on "reachability deficits" in quantum algorithms like QAOA [1, 9], and his emphasis on the need for careful analysis of noise and scalability in evaluating variational quantum algorithms, directly aligns with the motivations and findings of this paper. Uvarov et al.'s work provides a practical method for addressing the limitations posed by noise in VQAs, potentially improving their "reachability" and making them more viable for tackling complex problems. Moreover, Biamonte and collaborators have explored the phenomenon of "parameter concentration" in QAOA [13], where optimal parameters tend to cluster within narrow ranges. This concentration could amplify the impact of noise, as small errors in these sensitive parameters could significantly degrade performance, potentially limiting the effectiveness of Uvarov et al.'s ZNE method.**Hirota (2021):**Hirota's work on nonlinear errors in quantum processors [11] emphasizes the complex, interconnected nature of noise in quantum systems, suggesting that traditional error models based on independent single-qubit errors might be insufficient. Uvarov et al.'s reliance on a specific noise model, considering inhomogeneous error rates across qubit pairs, indirectly supports Hirota's perspective by acknowledging that error behavior can be dependent on the structure and interactions within the quantum circuit.

### References

[1] V. Akshay *et al.*, "Reachability deficits in quantum approximate optimization," Phys. Rev. Lett. **124**, 090504 (2020).

[9] J. Biamonte, "Universal variational quantum computation," Phys. Rev. A **103**, L030401 (2021).

[11] O. Hirota, "Introduction to semi-classical analysis for digital errors of qubit in quantum processor," Entropy **23**, 1577 (2021).

[13] V. Akshay *et al.*, "Parameter concentrations in quantum approximate optimization," Phys. Rev. A **104**, L010401 (2021).

### Conflict of Interest

The authors acknowledge support from research projects at the Skolkovo Institute of Science and Technology and the Russian Quantum Center, suggesting a potential bias towards promoting the capabilities of quantum computers. However, the paper also presents a critical analysis of noise limitations and acknowledges the need for further research on scalability, indicating a balanced perspective.