Technical Summary: Optimizing Symmetry Informed Probabilistic Error Cancellation

Problem Statement

Quantum Error Mitigation (QEM) techniques, such as Probabilistic Error Cancellation (PEC), are essential for extracting accurate results from noisy intermediate-scale quantum (NISQ) devices. However, PEC suffers from an exponential scaling of sampling costs with system size and error probability. While Quantum Error Detection (QED) can reduce these costs by post-selecting on symmetry measurements, the noise inherent in the symmetry measurement circuits themselves can negate the benefits of QED, potentially eliminating any advantage over PEC alone. The central problem addressed is how to optimally select symmetry measurement configurations to balance the reduction in undetectable errors against the introduction of noise from the measurement process, thereby minimizing the total mitigation overhead.

Methodology

The authors propose a framework for PEC+QED that integrates QED with PEC, treating the selection of symmetry measurements as a classical optimization problem.

1. Hybrid Mitigation Framework:

   • QED: Detects errors by measuring stabilizers (symmetries) and discarding runs where the measured eigenvalue deviates from the expected value.
   • PEC: Applied only to the remaining "undetectable" errors (those that commute with all measured stabilizers) to cancel their effect on average.
   • Cost Model: The total sampling cost is a product of the PEC overhead for undetectable errors and the QED overhead (inverse of the probability of no detection). Crucially, the noise introduced by the symmetry measurement circuits ($U_{SYM}$) is modeled as an additional cost that increases bias or variance.

2. Optimization Formulation:
   The authors cast the selection of stabilizers as a variant of the set covering problem. Given a set of candidate stabilizers $S$, the goal is to find a subset $C \subseteq S$ that minimizes a cost function $\zeta(C)$.

   • Objective: Minimize the sum of the cost of undetected errors (which determines the PEC overhead) and the cost of the symmetry measurements themselves.
   • Approaches:
     • Symmetry Circuit Mitigation: Assumes symmetry circuits can be mitigated to remove bias, minimizing the total quasi-probability cost. This is formulated as a constrained set covering problem where the benefit term is submodular, allowing for greedy approximation algorithms with performance guarantees.
     • Deferred Symmetry Measurements: Assumes symmetries are measured after the observable, so measurement noise does not bias the observable directly but increases variance via false detections. This formulation involves the difference of submodular functions, requiring heuristic optimization methods.
   • Error Propagation: The method accounts for how errors propagate through the circuit (specifically through Clifford and Pauli rotation layers) to determine which errors are detectable at the point of symmetry measurement.

3. Numerical Experiments:
   The framework was tested on two distinct quantum tasks:

   • GHZ State Preparation: Simulating $n$-qubit GHZ states using both linear-depth and logarithmic-depth circuits.
   • Fermi-Hubbard Model Dynamics: Simulating the time evolution of a spinless Fermi-Hubbard model on a square lattice using the Generalized Superfast Encoding (GSE).

Key Results

1. GHZ State Output Distributions

• Linear-Depth Circuits: For linear-depth circuits, standard local stabilizers (e.g., $Z_i Z_{i+1}$) were insufficient to provide an advantage over PEC alone because the noise cost of measuring them outweighed the reduction in PEC overhead. The optimization successfully identified that non-local stabilizers (e.g., $Z_0 Z_{n-1}$) were necessary to achieve a net benefit. With optimized non-local stabilizers, PEC+QED achieved a Total Square Error (TSE) reduction of nearly an order of magnitude compared to PEC for $n=50$.
• Logarithmic-Depth Circuits: These shallower circuits are less affected by idling noise, making the relative cost of symmetry measurements higher. Consequently, the advantage of PEC+QED over PEC was more restricted. While an advantage existed for $n \ge 15$, the empirical sampling costs were often comparable to PEC alone for larger $n$, highlighting the sensitivity of the approach to circuit structure and error propagation.

2. Fermi-Hubbard Model Time Dynamics

• Observable Estimation: For a $2 \times 2$ lattice, PEC+QED consistently outperformed both standalone PEC and QED in estimating Hamiltonian terms and vertex operator products. It achieved the lowest Root Mean Square Error (RMSE) across all evolution times.
• Correlators:
  • Two-point charge correlator: PEC+QED maintained the lowest RMSE, with performance degradation rates closer to QED than PEC as circuit depth increased.
  • Radial charge correlator: For values close to zero, PEC and PEC+QED performed comparably, suggesting the advantage is most pronounced for observables with larger signal magnitudes.
• Scalability and Partial Stabilizers: As the lattice size $m$ increased, the sampling overhead for all methods grew super-exponentially. However, the optimization framework revealed that for larger systems ($m=5, 6$), measuring only a subset of stabilizers (e.g., 78–83% of available loop operators) minimized the total sampling cost. This "partial" configuration reduced the overhead compared to measuring all stabilizers, with the savings increasing as the system size grew.

Significance and Claims

The paper claims that circuit-specific tailoring of QEM techniques is critical for achieving practical advantages. The key contributions are:

1. Optimization of Symmetry Usage: The authors demonstrate that blindly applying all available symmetries is suboptimal. By formulating the selection of stabilizers as a classical optimization problem, one can systematically suppress the impact of noisy symmetry measurements.
2. Context-Dependent Advantage: The advantage of PEC+QED is not universal; it depends heavily on the circuit structure (e.g., linear vs. logarithmic depth) and the distribution of detectable errors. In some cases (linear-depth GHZ), non-local stabilizers are essential; in others (Fermi-Hubbard), partial stabilizer sets are optimal.
3. Near-Term Value of Fault-Tolerant Principles: The results suggest that fault-tolerant design principles, such as using stabilizer codes and symmetry verification, can provide immediate value for near-term devices when combined with probabilistic error cancellation, provided the measurement configurations are carefully optimized.
4. Scalability: The ability to reduce mitigation overheads by measuring subsets of stabilizers in larger systems indicates a pathway to scaling these hybrid techniques to more complex simulations.

The authors conclude that while PEC+QED does not replace the need for full fault-tolerant quantum computation (FTQC), it offers a viable strategy to bridge the gap between current noisy hardware and practical quantum advantage, provided that the trade-offs between detection capability and measurement noise are rigorously managed through optimization.