Quantum Process Tomography: Characterizing Quantum Operations

Table of Contents

1. Introduction
2. What Is Quantum Process Tomography (QPT)?
3. Motivation and Applications
4. Quantum Operations and Quantum Channels
5. Kraus Representation and Superoperators
6. The Choi–Jamiołkowski Isomorphism
7. Process Matrix (Chi Matrix) Representation
8. Steps in Quantum Process Tomography
9. Choosing Input States
10. Performing Output State Tomography
11. Reconstructing the Process Matrix
12. Physical Constraints on the Process Matrix
13. Maximum Likelihood Estimation in QPT
14. Gate Set Tomography (GST)
15. Compressed and Scalable QPT
16. Ancilla-Assisted Process Tomography
17. Experimental Implementations
18. Common Errors and Mitigation Techniques
19. QPT vs Quantum State Tomography
20. Conclusion

1. Introduction

Quantum Process Tomography (QPT) is a method used to reconstruct and fully characterize the dynamics of a quantum operation or channel. It reveals how an input quantum state is transformed under a physical process.

2. What Is Quantum Process Tomography (QPT)?

QPT determines the complete description of a quantum process \( \mathcal{E} \) by measuring how it acts on a set of known input states. This allows benchmarking, verification, and error analysis of quantum gates and devices.

3. Motivation and Applications

• Characterize and verify quantum gates
• Benchmark quantum processors
• Detect and diagnose decoherence
• Validate quantum control schemes

4. Quantum Operations and Quantum Channels

A quantum process is mathematically described as a completely positive, trace-preserving (CPTP) map:

\[
ho \mapsto \mathcal{E}(
ho)
\]

It can be represented using operator-sum (Kraus), superoperator, or process matrix formulations.

5. Kraus Representation and Superoperators

Any quantum process can be expressed as:

\[
\mathcal{E}(
ho) = \sum_k E_k
ho E_k^\dagger
\]

where \( \{E_k\} \) are Kraus operators satisfying \( \sum_k E_k^\dagger E_k = I \).

6. The Choi–Jamiołkowski Isomorphism

This maps processes to density matrices via:

\[
\chi_{\mathcal{E}} = (\mathcal{E} \otimes \mathbb{I})(|\Phi^+
angle\langle\Phi^+|)
\]

This state \( \chi \) encodes all the information about \( \mathcal{E} \) and can be experimentally reconstructed.

7. Process Matrix (Chi Matrix) Representation

The process can also be written as:

\[
\mathcal{E}(
ho) = \sum_{m,n} \chi_{mn} E_m
ho E_n^\dagger
\]

where \( \{E_m\} \) are a fixed operator basis (e.g., Pauli operators) and \( \chi \) is the process matrix to be estimated.

8. Steps in Quantum Process Tomography

1. Prepare a complete set of input states \( \{
   ho_j\} \)
2. Apply the quantum process \( \mathcal{E} \)
3. Perform quantum state tomography on each output \( \mathcal{E}(
   ho_j) \)
4. Reconstruct the process matrix \( \chi \)

9. Choosing Input States

For a single qubit, a common set includes:

10. Performing Output State Tomography

Each output state is reconstructed using standard quantum state tomography techniques, measuring observables such as \( \sigma_x, \sigma_y, \sigma_z \).

11. Reconstructing the Process Matrix

Using the known transformation from each input state to output state, solve a linear system or perform maximum likelihood estimation to obtain the best-fit process matrix \( \chi \).

12. Physical Constraints on the Process Matrix

The reconstructed \( \chi \) must be:

• Hermitian
• Positive semidefinite (\( \chi \geq 0 \))
• Trace-preserving: \( \sum_{m,n} \chi_{mn} E_n^\dagger E_m = I \)

13. Maximum Likelihood Estimation in QPT

MLE improves physicality and robustness:

• Iteratively maximizes the likelihood of measured data
• Ensures physical \( \chi \)
• Avoids unphysical artifacts from noise

14. Gate Set Tomography (GST)

An advanced technique that:

• Characterizes the full gate set (preparation, gates, measurement)
• Eliminates SPAM (state-prep and measurement) errors
• Uses self-consistency instead of external calibration

15. Compressed and Scalable QPT

Methods to handle large systems:

• Compressed sensing: assumes sparsity
• Randomized benchmarking (RB): yields average fidelity
• Neural-network-based estimators

16. Ancilla-Assisted Process Tomography

Uses entangled ancilla states (e.g., Bell states) to directly probe the Choi matrix. Requires fewer resources and can access coherences more efficiently.

17. Experimental Implementations

QPT has been implemented in:

• Superconducting qubits
• Trapped ions
• NV centers
• Photonic systems
  Readout fidelities and calibration affect reconstruction accuracy.

18. Common Errors and Mitigation Techniques

• SPAM errors: mitigated by GST or error correction
• Shot noise: reduced by averaging
• Drift and instability: require real-time calibration

19. QPT vs Quantum State Tomography

Feature	QST	QPT
Reconstructs	Density matrix \( ho \)	Process map \( \mathcal{E} \)
Inputs needed	Single state	Multiple known states
Output	2ⁿ × 2ⁿ matrix	4ⁿ × 4ⁿ matrix

20. Conclusion

Quantum Process Tomography is vital for verifying and understanding quantum operations. While it scales poorly with system size, it remains an indispensable tool in quantum hardware development, often used alongside scalable alternatives like RB and GST.