Quantum Process Tomography

• Quantum process tomography is a method for completely characterizing quantum channels, capturing both unitary dynamics and noise effects.
• It employs techniques like ancilla-assistance, selective measurements, and tensor network compression to overcome exponential scaling challenges.
• Recent advances integrate machine learning and error-mitigation strategies to enhance SPAM correction and boost process fidelity.

Quantum process tomography (QPT) is the comprehensive procedure for reconstructing the full quantum channel or dynamical map governing the evolution of quantum states in an open or closed system. QPT yields a complete representation (typically in the form of a process matrix or a superoperator) that enables prediction of the output for any valid input state, encompassing both coherent (unitary) and incoherent (dissipative, decohering, or noise) processes. As an indispensable diagnostic tool, it underpins quantum hardware validation, error correction, and benchmarking, but traditional approaches face scalability and noise-related challenges. Contemporary advances in QPT address these hurdles through improved experimental designs, algorithmic developments, and integration with machine learning, making process identification tractable for increasing system sizes.

1. Foundational Concepts and Standard Methodology

QPT formally characterizes an unknown quantum operation $\mathcal{E}$ via its action on an operator basis. In the so-called operator-sum (Kraus) or $\chi$-matrix representation:

$$\mathcal{E}(\rho) = \sum_{mn} \chi_{mn} E_m \rho E_n^\dagger,$$

where $\{E_m\}$ forms a basis of operators (e.g., Pauli or Gell-Mann matrices) over the system Hilbert space, and $\chi$ is the process matrix fully encoding the channel's properties.

The canonical procedure for a $d$-dimensional system involves:

1. Preparing a complete basis of input states $\{\rho_j\}$,
2. Applying the unknown channel,
3. Performing quantum state tomography (QST) on each output state $\mathcal{E}(\rho_j)$ to reconstruct its density matrix, and
4. Solving a linear inversion to obtain $\chi$ (or equivalently the Choi matrix) by relating the measurement data to the process-action on the operator basis.

The fundamental bottleneck of this technique is exponential scaling: the number of required measurements and the size of the $\chi$-matrix both grow as $d^4$ for a $d$-dimensional system, thus rapidly becoming intractable for more than a few qubits (1105.4815, Shukla et al., 2014, Dang et al., 2021).

2. Scalability, Selectivity, and Resource-Efficient QPT

Addressing this scaling challenge, several methodologies depart from tomographic completeness in favor of selectivity or compressed representations:

• Ancilla-less Selective QPT: By mapping individual $\chi_{ab}$ elements to measurable average survival probabilities over a 2-design of pure states, specific elements can be estimated efficiently without reconstructing the entire process (1105.4815). Physical implementability of non-CP maps is circumvented via linear combinations of CP maps and careful input state preparation exploiting mutually unbiased bases or Clifford group transformations.
• Direct/Parallel QPT: Schemes based on weak measurements or single-shot readout exploit pointer observables and postselection to directly estimate process parameters, reducing both state preparation and measurement complexity (Zhang et al., 2013, Shukla et al., 2014). Notably, in direct and parallel weak measurement QPT, each parameter requires only five measured quantities, independent of Hilbert space size.
• Single-Shot QPT (SSPT and AAQST): By employing ancilla, all process information can be distributed into a system-ancilla state, allowing full process determination via a single collective measurement in a basis of commuting observables (Shukla et al., 2014). While this requires additional quantum resources (ancilla qubits), the reduction in measurement overhead is exponential.
• Tensor Network and Process Compression: Large-scale systems benefit from representing the process Choi matrix as a locally purified tensor network (LPDO), with contraction paths tailored to experimental topology, reducing classical computational cost by orders of magnitude (Dang et al., 2021).

3. Incorporating and Mitigating SPAM Errors

State preparation and measurement (SPAM) errors constitute a fundamental source of bias in traditional QPT. Recent protocols mitigate SPAM as follows:

• Self-Consistent QPT: By simultaneously estimating errors in all gates used for SPAM and process implementation, one eliminates the bias via likelihood functions modeling all sources of imperfection, with constraints enforced via semidefinite programming (Merkel et al., 2012).
• Calibration and Digital Twins: Calibration data from "process tomography on nothing" (idle/identity process) are used to construct a direct estimate of the SPAM error superoperator (Blume-Kohout et al., 20 Dec 2024). A further innovation is the use of machine learning—variational autoencoders trained on SPAM error data—to generate "digital twins" of error matrices, which statistically refine the probe operators for process estimation, providing order-of-magnitude improvements in process fidelity over standard QPT (Huang et al., 12 May 2025).
• Gauge Regularization and Least-Squares Correction: Explicit correction schemes adjust the measured process matrix via symmetric or asymmetric splitting of the SPAM error, ensuring the SPAM-corrected estimate remains as close as possible to the physical operations assumed in the experiment (Blume-Kohout et al., 20 Dec 2024).

4. Advanced Process Reconstruction and Learning-Based Approaches

The high-dimensional estimation problem of QPT is ameliorated using advanced algorithmic techniques:

• Gradient-Descent over Kraus or Choi Parameterizations: QPT is cast as the optimization of a parameterized set of Kraus operators, with trace-preserving and complete positivity constraints enforced via gradient descent on the Stiefel manifold; the approach is efficient for low-rank (or approximately low-rank) channels (Ahmed et al., 2022).
• Universal Compilation: Process characterization is reframed as the compilation of a set of Kraus operators or a Choi matrix, trained so that their action (augmented with a known inverse) restores arbitrary input states (Linh et al., 21 Apr 2025). Riemannian optimization on the Stiefel manifold undergirds the numerical implementation, and the method is validated on random unitaries and noise channels.
• Variational and Machine Learning QPT: The unknown process is encoded in a parametric quantum circuit (PQC) whose parameters are optimized such that the PQC output best matches the actual process as measured on a compact set of input–output pairs, yielding high-fidelity approximation with two orders of magnitude fewer measurements for $n\leq 8$ qubits (Xue et al., 2021).

5. Shadow Tomography, Sample Complexity, and Generalized Measurement Techniques

Recent advances employ randomized "classical shadows" and Choi isomorphism to achieve efficient process estimation:

• ShadowQPT: By mapping process tomography to quantum state shadow tomography via Jamiołkowski isomorphism and using random Pauli or Clifford measurements, simultaneous estimation of all $k$-qubit reduced processes is achieved with sample complexity logarithmic in $n$ (for fixed $k$) (Levy et al., 2021).
• Optimal POVM-SPT: Generalizing classical shadow tomography to optimize over POVMs, the shadow norm governing sample complexity is minimized via convex optimization. Simulated annealing is used to identify POVMs with lower shadow norm than unitary-based schemes, yielding up to $2^{180}$-fold reduction in sample complexity for 64-qubit systems (Wang et al., 30 Jun 2025). The combination of Choi state preparation and informationally complete, optimal POVMs defines the new benchmark for process sample efficiency.

Methodology	Key Features	Resource/Scaling Notes
Selective QPT (1105.4815)	2-design survival averages; no ancilla	Poly($n$) scaling, selective params
SSPT (Shukla et al., 2014)	Ancilla, one measurement, AAQST	Single-shot, exponential reduction
Gradient/PLS/Kraus (Surawy-Stepney et al., 2021, Ahmed et al., 2022, Linh et al., 21 Apr 2025)	Learning-based, convex, manifold opt.	Poly($n$) for low-rank, scalable
ShadowQPT/POVM-SPT (Levy et al., 2021, Wang et al., 30 Jun 2025)	Randomized, Choi-state, minimal samples	$\log(n)$ or $O(1)$ for $k$-local

6. Practical and Experimental Demonstrations

QPT protocols have been experimentally validated in several architectures:

• Photonic Qubits: Heralded single-photon and interferometric encodings are used to implement 2-qubit selective QPT, showing over 90% fidelity while requiring orders-of-magnitude fewer measurements (1105.4815).
• NMR and Trapped-Ion Platforms: Ancilla-assisted and single-shot QPT have been demonstrated for single- and two-qubit processes, including full χ-matrix reconstructions for gates such as CNOT and phase gates, and dissipative twirling processes (Shukla et al., 2014, Verdeil et al., 2023).
• Superconducting Circuits: Digital twin–assisted process tomography demonstrates at least an order-of-magnitude improvement in gate fidelity over standard QPT under engineered SPAM errors (Huang et al., 12 May 2025).
• Room-Temperature Alkali-Metal Vapors: QPT of a $^{87}$Rb ensemble yields high-fidelity Liouvillian reconstruction for qutrits using polarization rotation measurements and overcomplete input state sets, supporting open-system characterization and quantum sensing (Sun et al., 27 Aug 2025).
• Noisy NISQ Devices: Compressing time-evolution circuits and benchmarking via QPT reveals that variationally compressed circuits yield process channels with larger eigenvalue moduli—suggesting superior noise resilience compared to Trotterized implementations (Dinca et al., 29 Sep 2025).

7. Theoretical, Algorithmic, and Practical Impact

The evolution of QPT from complete tomographic reconstructions to resource-efficient, robust, and noise-adapted protocols enables the extension of full-channel characterization to larger devices and more complex processes. Selective, variational, and shadow-based approaches offer tractable paths for real-time process diagnostics on quantum processors, integrating error mitigation and SPAM correction directly into QPT workflows. The transition from operator inversions toward learning-based, manifold-constrained, and hybrid quantum–classical protocols marks a significant conceptual shift. As quantum platforms grow and applications such as real-time error tracking and on-the-fly process certification become paramount, these advances in QPT will be central to achieving truly scalable, robust quantum information processing.

In conclusion, quantum process tomography has progressed from an exponential-scaling, error-prone task into a robust, versatile toolbox, underpinning the entire spectrum of quantum device characterization—from local noise diagnosis to non-unitary open-system modeling (1105.4815, Merkel et al., 2012, Zhang et al., 2013, Zhou et al., 2014, Shukla et al., 2014, Deville et al., 2019, Surawy-Stepney et al., 2021, Xue et al., 2021, Leyton-Ortega et al., 2021, Levy et al., 2021, Dang et al., 2021, Ahmed et al., 2022, Verdeil et al., 2023, Deville et al., 18 Jul 2024, Blume-Kohout et al., 20 Dec 2024, Linh et al., 21 Apr 2025, Huang et al., 12 May 2025, Wang et al., 30 Jun 2025, Sun et al., 27 Aug 2025, Dinca et al., 29 Sep 2025).