# Techniques: Simplifying errors by symmetry and randomisation

## What the paper contributes (one paragraph)
The paper presents a set of quantum circuit parallelisation methods that reduce the complexity of the errors affecting a noisy computation, where error complexity is the number of distinct Pauli coefficients needed to describe a stochastic Pauli channel. By running the same circuit across different qubit subsets (or at different times) and combining the outputs, the individual error channels are averaged into a simpler effective channel. Two mechanisms are analysed: symmetrisation, where circuit and device-topology symmetries collapse equivalent Pauli coefficients, and randomisation, where averaging over independent channels converges towards a global depolarising channel or a convex mixture of depolarising channels. Reductions are shown to be constant, linear, or exponential in system size. The methods are demonstrated on superconducting hardware and in simulation as enhancements to existing error mitigation: lower sampling cost for matrix-inversion measurement error mitigation, better noise-estimation circuit mitigation, and more predictable circuit performance under time-dependent noise.

## Techniques offered
- **error symmetrisation by quantum circuit parallelisation** - runs a circuit under symmetry-equivalent qubit mappings and combines the outputs so that Pauli coefficients related by a device or circuit symmetry become identical, yielding a lower-complexity effective stochastic Pauli channel that an external mitigation routine can characterise more cheaply.
  - guarantee: deterministic reduction in error complexity (number of distinct Pauli coefficients) by the symmetry group order, up to O(4^n/2n) or O(n^3/6) for permutation symmetry, under the stated gate-independence assumptions; no statistical soundness claim beyond this.
  - quote: "For the symmetry reductions, coefficients of operators that can be mapped to each other by a given symmetry transformation become the same in the effective error channel."
- **error randomisation by quantum circuit parallelisation** - averages independent stochastic Pauli channels from parallel circuits so the effective channel converges on a global depolarising channel (model (r,1)) or a convex combination of depolarising channels (model (r,2)), supplying near-depolarising noise that downstream techniques can assume.
  - guarantee: convergence of the effective channel to error complexity O(1) under error model (r,1) and O(2^n) under (r,2) as the number of parallel instances grows; convergence holds under the stated i.i.d. coefficient assumptions, no per-shot bound.
  - quote: "In other words, the effective error channel converges on a global depolarizing channel."
- **sample-complexity reduction for matrix-inversion measurement error mitigation** - applies the symmetry reduction so that only the symmetry-inequivalent basis-state response distributions need be characterised, cutting the calibration sampling overhead of an external measurement error mitigation result without degrading its effectiveness.
  - guarantee: empirical statistical bound; on hardware the symmetrised calibration used 60,000 samples versus 160,000 for unaided mitigation while achieving approximately the same expectation-value and total-variation-distance error reduction.
  - quote: "The first is applying symmetry error simplification to reduce the sample cost of measurement error mitigation."

## Where it could apply
- Target primitive(s): VQE and other variational expectation-value circuits, sampling, and any Clifford-plus-single-qubit-rotation circuit whose output is measured in the computational basis.
- Target application group(s): error_correction (noise tailoring and mitigation), platforms_tooling (benchmarking and characterisation, measurement error mitigation, noise-estimation circuit mitigation).
- Code family / hardware assumptions: not a fault-tolerant code; physical-layer NISQ devices with stochastic Pauli (or bit-flip) noise, demonstrated on superconducting hardware, assumes randomised compiling to enforce the stochastic Pauli noise form and a symmetric device topology for the symmetry reductions.

## Caveats
The techniques are noise-simplification and mitigation-enhancement methods rather than a code, decoder, or verification protocol; they supply lower-complexity effective error channels and reduced mitigation sampling cost to external mitigation, characterisation, and benchmarking results. They are not stand-alone error correction and provide no fault-tolerance or interactive-soundness guarantee. Symmetry reductions rely on local-gate-independent, multiqubit-gate-dependent noise and on device-topology symmetry; randomisation reductions rely on i.i.d. assumptions across parallel circuits and on randomised compiling to enforce stochastic Pauli noise. Parallelisation consumes additional qubits or repeated time slots, and the demonstrations are small (four-qubit hardware, ten-circuit simulation).