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CHAPTER 25

# Integrated laboratories and a path to research

Learning goals. Combine mathematical derivation, simulation, statistical inference, and evidence assessment in reproducible projects.

## 25.1 Project 1: identify a preparation

A source produces one of three preparations: |+⟩|+\rangle, |+i⟩|+i\rangle, or an equal classical mixture of |0⟩|0\rangle and |1⟩|1\rangle. Design a measurement plan to distinguish them statistically.

Predict X,Y,Z distributions before sampling. All three have balanced Z outcomes. In X, |+⟩|+\rangle is deterministic while the other two are balanced. In Y, |+i⟩|+i\rangle is deterministic while the other two are balanced. Thus X and Y data distinguish the three preparations; Z alone cannot.

Use L02/L03 for the pure states and L11 with mixing set to one for the maximally mixed state. Choose a shot budget and explain how sampling uncertainty affects a classification. For a general unknown source, do not interpret an observed all-plus run as mathematical proof of a pure state.

Deliverable. A preparation table, exact projection calculations, simulated counts with seeds, a decision rule, and a discussion of its finite-data errors. A strong report distinguishes a model-selection exercise among three promised candidates from full tomography of an arbitrary state.

## 25.2 Project 2: an entanglement claim under noise

Use L07 to prepare a Bell state and inspect its local reductions. Use L08’s noisy Bell model to scan visibility v and CHSH settings. Derive the optimal S value 2 2 v 2\sqrt2v and threshold v>1/2 v>1/\sqrt2 before sampling.

Compare the Bell state with a classically correlated mixture of 00 and 11. Both have local state I/2 and perfect Z correlations, but they differ in other correlations. For the classical mixture, ⟨X X⟩=0\langle XX\rangle=0 and ⟨Z Z⟩=1\langle ZZ\rangle=1; for |Φ+⟩|\Phi^+\rangle, both are one. Local entropy alone does not certify entanglement.

Repeat at small and large shot counts near the violation boundary. Quantify uncertainty rather than reporting only whether one sampled S happened to exceed two. The simulator assumes the specified joint distribution; it cannot test experimental loopholes.

Deliverable. Density matrices, reduced states, a correlation table, S-versus-visibility analysis, and an explanation of which statements are properties of the model and which would require a physical experiment.

## 25.3 Project 3: recover a hidden structure

Choose either Simon’s algorithm or small-instance order finding. Record the oracle’s promise separately from the information supplied to the classical reconstruction.

For Simon, generate measured rows until rank n−1 is reached. Show binary elimination explicitly. Repeat with fewer samples and explain why multiple nonzero candidates remain. For order finding, sample several Fourier outcomes and show which convergents pass the modular-order test and produce nontrivial gcd factors.

A failed sample is an expected part of these algorithms. Do not delete it from the report. Estimate the number of complete repetitions required for a target confidence, and include the oracle implementation in any runtime discussion.

Deliverable. An exact small-instance distribution, raw sampled results, reconstruction steps, verified answers, and a cost-model comparison with a relevant classical method. The visible educational oracle table should not be presented as a real hidden-input benchmark.

## 25.4 Project 4: choose a code under a resource budget

Begin with the repetition-code model and compare d=3,5,7 under independent X errors. Explain why its impressive suppression does not protect arbitrary quantum states. Then use the distance-three surface-code lab to identify a single error, a stabilizer-equivalent recovery, and a zero-syndrome logical string.

Use L25 to design an illustrative memory/computation budget. State A A, p p, p t h p_{\mathrm{th}}, L L, ϵ\epsilon, the chosen odd d, data patches, additional patches, cycle time, and logical depth. Verify the distance choice by direct substitution. Vary p by a factor of two and compare costs.

Deliverable. A code-scope comparison, syndrome calculations, a distance derivation, resource tables, and at least three omitted engineering costs. Do not label the result a prediction for a named machine without a calibrated model.

## 25.5 Project 5: a reproducible software comparison

Run the supplied Qiskit examples and reproduce an asymmetric three-qubit circuit in L06. Map book wire j to Qiskit wire n−1−j, and compare exact amplitudes up to global phase. Compare measurement probabilities rather than expecting identical seeded counts from different random-number generators.

Add a controlled gate and a basis measurement. Introduce an intentional ordering error, predict its effect, and show how an asymmetric test detects it. Then compare a single-qubit channel’s density matrix with the explicit Kraus calculation in Chapter 8.

Deliverable. Source circuits, environment versions, mathematical predictions, exact numerical comparisons with tolerances, finite-shot results, and a documented failure case. The objective is to build a debugging method, not simply collect matching screenshots.

## 25.6 Project 6: audit a current claim

Choose one primary paper from Chapter 24. Read its abstract, methods available to you, stated metric, and limitations. Create a claim table with columns for the statement, evidence type, measured or assumed quantities, comparison baseline, and the strongest conclusion justified.

If access is limited to an abstract, state that limitation and do not infer unreported details. Check publication and correction records. Separate the experiment’s result from architecture proposals or roadmaps discussed around it.

Deliverable. A short evidence-backed technical assessment. It should identify one result that is directly established, one extrapolation, and one useful next experiment or analysis. This exercise develops scientific reading, not a verdict about a company’s worth.

## 25.7 Assessing your work

Use four criteria: correctness of the mathematical model; traceability from inputs to computed outputs; appropriate interpretation of uncertainty and assumptions; and clarity of the connection to the scientific question. A polished figure cannot compensate for a wrong basis convention. A correct formula cannot compensate for a claim about a different experiment.

For numerical work, include checks that could fail: probability normalization, channel trace preservation, known-state outputs, inverse recovery, and independent comparisons. State the tolerance and why it is sensible. Tests are strongest when they check an independent property rather than reimplementing the same expression.

For open-ended work, a negative or qualified conclusion can be excellent. A model that does not support a proposed advantage has still taught something if the analysis is transparent.

## 25.8 Where to go next

For information theory, study completely positive maps, entropy inequalities, channel capacities, and entanglement theory [4]. For algorithms, study query lower bounds, Hamiltonian simulation, block encodings, and quantum signal processing [23]. For error correction, study stabilizer formalism, decoding, fault-tolerant gadgets, and resource estimates [33, 37].

For hardware, choose a physical platform and learn its Hamiltonian, control methods, noise characterization, and fabrication or optical constraints. For software, contribute a small reproducible benchmark, simulator feature, or verified circuit transformation. The ability to connect assumptions, equations, code, and evidence is a useful research skill across all of these paths.

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