A Practical Guide to Running the SUPERPOSITION Benchmark on Your QPUQuantum hardware performance is still evolving rapidly, and benchmarks are essential tools for comparing devices, diagnosing problems, and guiding system improvements. The SUPERPOSITION Benchmark specifically targets a core quantum computing capability: the reliable preparation, manipulation, and measurement of large-scale superposition states. This guide walks you through the principles behind the benchmark, preparation steps, practical implementation on common QPU architectures, interpreting results, troubleshooting, and ways to use those results to optimize your device or experiments.
What the SUPERPOSITION Benchmark Measures
The SUPERPOSITION Benchmark assesses how effectively a quantum processing unit (QPU) can create and preserve coherent superposition across multiple qubits, execute circuits that manipulate those superpositions, and report accurate measurement outcomes. Key properties it probes include:
- State preparation fidelity: How close the prepared multi-qubit state is to the ideal equal superposition across computational basis states.
- Coherence and decoherence effects: How well phase relationships and amplitude balance persist as circuits grow in depth or qubit count.
- Gate and readout errors: How single- and multi-qubit gate infidelities and measurement errors degrade the expected probability distribution.
- Cross-talk and correlated errors: Whether operations on subsets of qubits disturb others, creating nontrivial correlations that deviate from ideal behavior.
- Scalability: How performance trends as you scale qubit count and circuit depth.
Why Run the SUPERPOSITION Benchmark
- Validate hardware readiness for algorithms that rely on global superpositions (e.g., quantum Fourier transform steps, certain variational forms, or sampling tasks).
- Compare different QPUs, qubit topologies, or firmware/compilation strategies.
- Identify noise sources that disproportionately affect large-scale state preparation.
- Track performance regressions after updates to control electronics, calibration routines, or qubit fabrication.
High-Level Benchmark Design
A typical SUPERPOSITION Benchmark does the following:
- Select N qubits (contiguous or by a chosen topology).
- Initialize all qubits to |0>.
- Apply gates to produce an equal superposition across the chosen computational subspace—commonly Hadamard gates on each qubit to create (1/sqrt(2^N)) Σ_x |x>.
- Optionally apply layers that test phase stability, entangling gates, or randomized compilations around the superposition.
- Perform measurements in the computational basis (Z-basis) and possibly other bases to probe coherence.
- Repeat shots and collect counts to form empirical probability distributions.
- Compute metrics comparing empirical results to the ideal distribution.
Common metrics:
- Hellinger distance or total variation distance (TVD) between empirical and ideal distributions.
- Fidelity estimates via linear inversion or maximum-likelihood state estimation for small N.
- Entropy measures and Kullback–Leibler divergence for distributional shifts.
- Statistical tests for correlated errors and cross-talk (e.g., mutual information between measurement outcomes of qubit subsets).
Practical Steps: Before You Run
- Choose qubit set and mapping: Prefer qubits with higher T1/T2 and lower gate/readout error rates. For large-N runs, decide whether to use contiguous physical qubits or a topology-aware subset.
- Calibrate: Ensure single-qubit gates, two-qubit gates (if used), and readout calibrations are fresh.
- Shot budget: Determine how many measurement shots you’ll collect per circuit—typically 2k–10k shots for stable statistics for moderate N. Larger N increases required shots exponentially if you want to resolve rare basis states.
- Compilation choices: Select native-gate decompositions and transpilation options that minimize added depth, especially extra two-qubit gates. Use error-aware routing if available.
- Randomization and seeds: If using randomized compilations or phase-perturbation layers, fix random seeds for reproducibility.
Example Implementations
Below are concise examples for three common hardware paradigms. Replace pseudocode and API calls with your QPU vendor’s SDK (IBM Qiskit, Rigetti Forest, Quantinuum, IonQ, or others).
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Superconducting (IBM-style, Qiskit-like)
from qiskit import QuantumCircuit, transpile, assemble, Aer, execute # Choose qubits and create circuit qc = QuantumCircuit(n) for q in range(n): qc.h(q) # Optional: add entangling layer to stress cross-talk # qc.cz(0,1); ... qc.measure_all() tqc = transpile(qc, backend=backend, optimization_level=1) job = backend.run(assemble(tqc, shots=5000)) counts = job.result().get_counts() # Compute TVD or Hellinger distance vs ideal uniform distribution -
Trapped ions (IonQ/Quantinuum style — native multi-qubit gates may differ)
qc = Program() # pseudocode qc.apply_global_hadamards(qubits) # Optionally insert phase-insensitive operations or wait times qc.measure_all() result = run_on_hardware(qc, shots=2000) counts = result.counts -
Photonic / Boson-sampling-like setups (specialized readout and state prep)
- Implement the optical equivalent of a uniform superposition (beam-splitter networks), collect samples, and map detection patterns to basis states.
- Use vendor-specific APIs for configuration and data retrieval.
Metrics: How to Compute & Interpret
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Ideal probability for each of 2^N basis states = ⁄2^N. For measured counts c_x over S shots, empirical p_x = c_x / S.
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Total Variation Distance (TVD): TVD = 0.5 * Σ_x |p_x − ⁄2^N|
- TVD near 0: close to ideal uniform distribution.
- Higher TVD: indicates bias, noise, or loss of superposition.
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Hellinger distance H: H = (1/√2) * sqrt(Σ_x (sqrt(p_x) − sqrt(⁄2^N))^2)
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Fidelity estimate for small N via state tomography (costly for larger N).
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Correlation analysis:
- Compute pairwise mutual information I(i;j) across qubits to detect cross-talk.
- Test for parity biases or systematic zero/one skews.
Interpret trends:
- TVD rising with N → scaling-limited coherence or compilation overhead.
- TVD rising with depth → gate errors or decoherence during circuit execution.
- Specific bit biases → readout calibration errors or decay during measurement window.
- Nonlocal correlation structure → cross-talk or correlated noise sources.
Troubleshooting Common Problems
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Strong bias toward |0…0>:
- Check T1 relaxation and idle times; reduce circuit duration or insert dynamical decoupling if supported.
- Verify readout assignment errors; run readout calibration and confusion matrix checks.
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High variation across runs:
- Ensure stable calibration, run benchmark multiple times to estimate variance.
- Control for temperature/drift in cryogenic systems; stagger runs to sample different calibration cycles.
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Unexpected correlations:
- Run isolated single-qubit superpositions to profile individual behavior.
- Inject isolation tests (idle gates on neighbors) to detect cross-talk.
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Rapidly increasing TVD with depth:
- Re-optimize transpiler to reduce two-qubit gate counts.
- Use randomized compiling to convert coherent errors into stochastic errors and average them out.
Reporting and Comparisons
When publishing or comparing results, include:
- QPU identifier, topology, and qubit indices used.
- Calibration snapshot: T1, T2, single- and two-qubit gate error rates, and readout error rates.
- Transpiler and gate set used.
- Circuit depth, number of shots, and random seeds.
- Raw counts and computed metrics (TVD, Hellinger, mutual information).
- Confidence intervals or standard errors for metrics across repeated runs.
A simple reporting table:
| Item | Value |
|---|---|
| QPU | Device name |
| Qubits used | list |
| N | number of qubits |
| Shots | 5000 |
| TVD | 0.12 ± 0.01 |
| Hellinger | 0.09 |
| Notes | entangling layer added / idle time = X μs |
Advanced Variants & Extensions
- Phase-coherence probe: After generating superposition, apply a global phase-sensitive operation (e.g., a multi-qubit rotation or QFT-like layer) and measure in X/Y bases to probe phase decay.
- Randomized compiling: Wrap superposition circuits with random Pauli/frame changes to average coherent errors.
- Subspace-specific superpositions: Prepare uniform superpositions over subsets of basis states to test targeted state preparation (useful for algorithmic subroutines).
- Error mitigation: Use readout error mitigation, zero-noise extrapolation, or other techniques to estimate underlying ideal distribution.
Practical Example: Analysis Workflow (small N)
- Generate and run H^⊗N circuit, collect counts for S shots.
- Compute p_x and TVD. If TVD > threshold (e.g., 0.1), inspect per-bit marginals and confusion matrix.
- Run single-qubit superposition and two-qubit benchmarks to isolate whether errors are local or correlated.
- Apply randomized compiling variant and see if TVD reduces (indicates coherent error contribution).
- If improvements are found, iterate on compilation and calibration.
Final Notes
The SUPERPOSITION Benchmark is a practical, scalable probe of a QPU’s ability to create and maintain delocalized quantum states. It’s straightforward to implement but reveals a wide range of hardware behaviors — from single-qubit readout bias to large-scale correlated noise — that matter for algorithmic performance. Use consistent reporting, control experiments, and repeated trials to draw meaningful conclusions about device capability and changes over time.
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