Quantum process tomography reconstructs a transformation rather than a state. The object to be estimated is a quantum channel, and the reconstruction must satisfy stronger physical constraints than ordinary matrix fitting: complete positivity and trace preservation.
State tomography asks, “What density matrix best explains these measurements?” Process tomography asks, “What map best explains how many known inputs were transformed into many measured outputs?” The second task compounds state-preparation and measurement uncertainty with channel estimation.
A finite-dimensional quantum channel can be represented in several equivalent ways: Kraus operators, a superoperator matrix, a Pauli transfer matrix, a process matrix in a chosen operator basis, or a Choi matrix. Each representation makes different constraints and calculations convenient.
Prepare spanning inputs → apply unknown process → perform output tomography → reconstruct a linear map → enforce CPTP constraints → validate on new inputs or sequences.
1. A quantum channel is a linear map on density matrices
Write the channel as 𝓔. Its input-output rule is
ρ_out=𝓔(ρ_in).
Physical channels are linear, completely positive and trace preserving. Complete positivity ensures that 𝓔 remains positive even when applied to one part of an entangled state. Trace preservation keeps total probability equal to one.
Earlier in the series, channels were represented as 𝓔(ρ)=Σ_kK_kρK_k† with Σ_kK_k†K_k=I. Process tomography estimates the map from experimental data without assuming the Kraus operators are known.
2. Linear maps become matrices after vectorisation
A d×d density matrix has d² entries. If we stack its columns into a d²-dimensional vector, a linear channel becomes a d²×d² matrix S satisfying
vec(𝓔(ρ))=S vec(ρ).
This superoperator representation turns process tomography into linear system identification on operator space.
The matrix depends on the vectorisation convention. Column-stacking and row-stacking differ by permutations. A process reconstructed under one convention can look transposed or reshuffled under another while representing the same physical channel.
3. Pauli transfer matrix for a qubit
Use the operator basis {I,X,Y,Z}. Define transfer coefficients
R_{ij}=(1/2)Tr[P_i 𝓔(P_j)].
Then the four-component Pauli-coordinate vector transforms linearly by R.
For a trace-preserving qubit channel, the first row is fixed in the standard normalised convention. The remaining affine action on the Bloch vector can be written
r_out=M r_in+t.
Unital channels have t=0. Amplitude damping has nonzero t because it pulls the Bloch ball toward the ground-state pole.
4. Worked example: depolarising channel in Pauli coordinates
Take the qubit depolarising model
𝓔(ρ)=(1-p)ρ+pI/2.
In Bloch form, r→(1-p)r. Thus the Pauli transfer matrix scales the X, Y and Z coordinates by 1−p while preserving the identity coordinate.
If process tomography estimates shrink factors 0.93, 0.92 and 0.94 along X,Y,Z, a simple isotropic depolarising fit might estimate a common factor near 0.93. But the unequal observed values may indicate anisotropic noise or finite-sample fluctuation. Model reduction should be tested, not imposed automatically.
5. The Choi matrix
Let |Φ⟩=(1/√d)Σ_j|j⟩|j⟩ be a maximally entangled state. Apply the channel to one half:
J_norm=(I⊗𝓔)(|Φ⟩⟨Φ|).
This normalised operator has trace one for a trace-preserving channel. Another common convention multiplies by d, producing an unnormalised Choi matrix J with trace d.
The convention must be stated because trace-preservation constraints and fidelity formulas differ by factors of d.
6. Choi positivity is complete positivity
Choi’s theorem gives a decisive equivalence: a linear map is completely positive exactly when its Choi matrix is positive semidefinite.
Thus a property that originally quantified positivity on every possible reference extension becomes one finite matrix constraint.
This is why Choi matrices are so useful in channel reconstruction. Complete positivity becomes ordinary eigenvalue positivity of one matrix.
7. Trace preservation becomes a partial-trace constraint
Under the unnormalised Choi convention J=Σ_{jk}|j⟩⟨k|⊗𝓔(|j⟩⟨k|), trace preservation requires
Tr_output J=I_input.
Under the normalised maximally-entangled convention, the corresponding partial trace is I/d.
Again, the mathematics is simple once the convention is fixed, but factor mistakes can silently make a correct process look non-trace-preserving.
8. Worked identity-channel Choi state
For the identity channel, applying 𝓘 to half of |Φ⟩ changes nothing. Therefore the normalised Choi state is simply
J_norm=|Φ⟩⟨Φ|.
For a qubit, this is the Bell state |Φ+⟩. The process matrix of an ideal identity gate is therefore pure in Choi-state language.
A noisy channel generally produces a mixed Choi state. Channel comparison can then be expressed using state-fidelity-like quantities on Choi matrices, with appropriate normalisation conventions.
9. Standard process tomography uses a spanning input set
A d-dimensional channel acts on a d²-dimensional operator space. To determine it by input-output experiments, prepare enough linearly independent input density operators to span that space.
For a qubit, common choices include |0⟩, |1⟩, |+⟩ and |+i⟩. Their density matrices span I, X, Y and Z components.
Perform state tomography on each output. The collection of input-output pairs determines the channel linearly in the ideal noiseless model.
10. Worked qubit input set
The four input states have Bloch vectors
- |0⟩: (0,0,1)
- |1⟩: (0,0,−1)
- |+⟩: (1,0,0)
- |+i⟩: (0,1,0)
If their measured output Bloch vectors are known, solve for the affine map r_out=Mr+t. The average of the |0⟩ and |1⟩ outputs reveals t, while differences determine the Z column of M. The |+⟩ and |+i⟩ outputs give the X and Y columns.
This is process tomography written as ordinary affine geometry on the Bloch ball.
11. Linear inversion can violate complete positivity
Just as state-tomography frequencies can yield a non-positive density matrix, independently reconstructed output states can yield a superoperator whose Choi matrix has negative eigenvalues.
The measured equations may be satisfied exactly by a non-CPTP linear map because finite-sample and calibration inconsistencies do not necessarily correspond to any physical channel.
A raw linear inversion is useful diagnostically but should not automatically be reported as a physical quantum process.
12. Constrained channel reconstruction
A physical reconstruction can optimise a likelihood or least-squares objective over Choi matrices satisfying
J≥0- the appropriate trace-preserving partial-trace constraint
This is a convex feasible set in Choi space. Semidefinite programming can therefore be used for several channel-estimation objectives.
The result is physical by construction, but statistical uncertainty and measurement-model error remain.
13. Process tomography inherits SPAM bias
Suppose the intended input is |0⟩ but the preparation actually produces a slightly excited mixed state. If the reconstruction assumes perfect |0⟩ preparation, the unknown channel can absorb that discrepancy.
Likewise, readout bias can be misattributed to the channel. Standard process tomography therefore does not cleanly separate state preparation, gate action and measurement error.
This is one motivation for gate-set tomography, which estimates a self-consistent set of states, gates and measurements together.
14. Gauge freedom in self-consistent gate-set models
If states, operations and measurements are all unknown, some simultaneous transformations of their coordinate representations leave every observable sequence probability unchanged.
This is a gauge freedom. Individual matrix entries of a gate-set representation may change while all predicted experimental probabilities remain the same.
Physical interpretation should therefore focus on gauge-invariant predictions or use a declared gauge-fixing convention when comparing matrices.
15. Ancilla-assisted process tomography
Because the Choi matrix is obtained by applying the channel to half of a maximally entangled state, one can reconstruct the process by preparing that entangled input, applying the channel once to one subsystem and performing joint state tomography on the output pair.
This ancilla-assisted picture replaces multiple distinct system inputs with one larger entangled input state.
The resource trade-off changes rather than disappears: extra ancilla preparation, entangling fidelity and larger joint-state tomography are required.
16. Process fidelity and average gate fidelity
For a target unitary channel U and an estimated channel 𝓔, one can compare their normalised Choi states. The entanglement fidelity is closely related to Choi-state overlap.
For trace-preserving channels on d dimensions, average gate fidelity and entanglement fidelity satisfy
F_avg=(dF_e+1)/(d+1).
This relation is useful but requires compatible fidelity conventions. Squared versus unsquared state fidelity and normalised versus unnormalised Choi matrices must not be mixed.
17. Average fidelity is not a worst-case guarantee
A channel can have high average gate fidelity while containing a coherent error direction that accumulates badly in certain circuits.
Diamond-norm distance provides a worst-case operational metric over inputs possibly entangled with a reference, but it is harder to estimate experimentally.
No single scalar captures every relevant feature of a process. Benchmarking should match the application’s sensitivity to coherent, stochastic, leakage and correlated errors.
18. Channel unitality as a diagnostic
A channel is unital if 𝓔(I)=I. In Bloch-affine form this means t=0.
Depolarising and dephasing channels are unital in their standard forms. Amplitude damping is not.
If an estimated channel has a large translation vector, fitting it with a purely depolarising noise model can miss a systematic relaxation component.
19. Leakage complicates the model dimension
A physical qubit may occupy levels outside the computational subspace. Standard two-dimensional process tomography can then interpret leakage as apparent trace loss, state-dependent noise or non-CPTP behaviour if the reduced model is inappropriate.
One response is to enlarge the Hilbert-space model. Another is to treat leakage probability as a separate measured quantity.
A reconstruction cannot faithfully describe dynamics that lie outside its assumed state space.
20. Time dependence and drift
Process tomography assumes the same channel generated all the collected data. If calibration drifts during the experiment, the fitted result may represent an average of several different channels.
Randomising measurement order, interleaving calibration checks and fitting time-dependent models can help diagnose drift.
A beautifully physical Choi matrix can still be a poor description of any one moment if the underlying process was nonstationary.
21. Predictive validation
After reconstructing a process, test it on input states or gate sequences not used directly in the fit. Compare predicted output observables with independent measurements.
If single-use predictions succeed but repeated-gate sequences fail, the discrepancy may reveal context dependence, temporal correlations or coherent accumulation not captured by the assumed Markovian channel.
Process tomography should therefore be validated at the scale at which the model will be used.
22. Common misconception: a process matrix is unique without a basis convention
The physical map is basis independent, but its matrix coordinates are not. A χ matrix in one operator basis is not numerically identical to a Pauli transfer matrix or column-vectorised superoperator.
23. Common misconception: process tomography cleanly measures gate error alone
Standard process tomography is sensitive to preparation and measurement error. Without independent calibration or self-consistent modelling, those effects can contaminate the inferred channel.
24. Common misconception: a high process fidelity proves every circuit will work well
Average metrics can hide structured coherent errors, leakage and time correlations. Long circuits can amplify small coherent misrotations much more strongly than an average one-gate fidelity suggests.
25. Worked synthesis problem
Suppose a qubit channel maps Bloch vectors as
(x,y,z)→(0.9x,0.9y,0.8z+0.2).
Step 1: Identify unitality. The translation t=(0,0,0.2) is nonzero, so the channel is not unital.
Step 2: Ground-state fixed point. The equation z=0.8z+0.2 gives z=1, suggesting relaxation toward |0⟩.
Step 3: Compare transverse and longitudinal shrinkage. X and Y shrink by 0.9 while deviations in Z shrink by 0.8, so the noise is anisotropic.
Step 4: Model interpretation. This resembles amplitude damping with additional transverse decoherence rather than pure depolarising noise.
Step 5: Validation. Predict outputs for |+⟩ and |1⟩ and compare with held-out tomography data before accepting the fitted channel as stationary and Markovian.
26. Practice set
- What three properties define a finite-dimensional quantum channel?
- What is a superoperator matrix?
- What does the Pauli transfer matrix act on?
- How is a Choi matrix constructed?
- What Choi condition is equivalent to complete positivity?
- What partial-trace condition expresses trace preservation?
- Why can linear process inversion be nonphysical?
- What is SPAM contamination?
- What is the advantage of ancilla-assisted process tomography?
- Why is average gate fidelity not a worst-case guarantee?
- What does nonunitality look like in Bloch-affine form?
- Give one predictive validation test for a reconstructed channel.
Answers
- Linearity, complete positivity and trace preservation.
- A matrix representing the linear action of the channel after vectorising operators.
- Pauli-basis operator coordinates or the equivalent Bloch-affine representation for qubits.
- Apply the channel to one half of a maximally entangled state, with a declared normalisation convention.
- Positive semidefiniteness.
- The appropriate input identity under partial trace, scaled according to Choi convention.
- Finite-sample inconsistencies can produce a map whose Choi matrix has negative eigenvalues.
- State-preparation and measurement errors contaminating the inferred gate process.
- One entangled input can encode the full process into a larger output state.
- It averages performance and can hide coherent or worst-case directions.
- A nonzero translation vector t.
- Predict outputs or sequence statistics for data not used in the reconstruction.
Sources and further study
[1] I. L. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box. An early process-tomography formulation.
[2] J. F. Poyatos, J. I. Cirac and P. Zoller, Complete characterization of a quantum process. A foundational proposal for experimental process characterisation.
[3] J. M. Chow and colleagues, Randomized benchmarking and process tomography for gate errors in a solid-state qubit. An experimental comparison illustrating how process tomography and benchmarking answer related but different questions.
Continue through Quantum Mathematics
Guide 21: Quantum State Tomography, Informational Completeness and Physical Reconstruction reconstructs states. Guide 23: Classical Shadows, Randomised Measurements and Observable Prediction targets many observables without full process or state reconstruction. Guide 24: Randomised Benchmarking, Gate Fidelity and Error Budgets estimates operational gate performance from random sequences.