Advancing quantum process tomography through quantum compilation

Quantum process tomography (QPT) plays a central role in characterizing quantum gates and circuits, diagnosing quantum devices, calibrating hardware, and supporting quantum error correction. However, conventional QPT methods face challenges related to scalability and sensitivity to noise. In this work, we propose a QPT framework based on quantum compilation, which represents quantum processes using optimized Kraus operators and Choi matrices. By formulating QPT as a compilation and optimization problem, our approach significantly reducing measurement and computational overhead while maintaining reconstruction accuracy. We benchmark the method using numerical simulations of Haar-random unitary gates and demonstrate a reliable process reconstruction. We further apply the framework to dephasing channels with both time-homogeneous and time-inhomogeneous noise, as well as to depolarizing and amplitude-damping channels, where stable performance is observed across different noise regimes. These results indicate that quantum compilation-based QPT can serve as a practical alternative to standard QPT methods for quantum process characterization and device validation.

💡 Research Summary

The manuscript introduces a novel framework for quantum process tomography (QPT) that leverages quantum compilation techniques, termed Compilation‑based Quantum Process Tomography (CQPT). Traditional QPT requires preparing a complete set of input states, applying the unknown quantum channel, and performing full quantum state tomography on the outputs. This approach scales exponentially with the number of qubits, both in the number of required measurements and in the classical post‑processing cost, making it impractical for medium‑scale quantum devices.

CQPT reframes the tomography problem as a compilation task. The unknown channel 𝔈 is parameterized either by a set of Kraus operators k = (K₁,…,K_k)ᵀ or by its Choi matrix J_𝔈. The key idea is to treat these parameters as trainable “gate” objects that can be optimized so that, after applying the trainable channel followed by an (approximate) inverse of 𝔈, the initial state is recovered. Two theorems formalize the necessary conditions for exact reconstruction. Theorem 1 states that for a Kraus‑based representation the equality ρ_f ≡ 𝔈⁻¹(∑_l K_l ρ_in K_l†) = ρ_in must hold; this is exact for unitary or near‑unitary processes where an inverse exists or can be approximated linearly. Theorem 2 provides the analogous condition for the Choi representation, involving the pseudoinverse J_𝔈⁺ and a partial trace over the ancillary space, thereby covering irreversible channels such as dephasing or amplitude‑damping.

Both representations employ a cost function based on fidelity (or infidelity) between the recovered state and the original. Importantly, the cost can be evaluated using a single‑shot measurement: after preparing a random pure state |ψ⟩ = W|0⟩ (with W drawn from a finite Haar‑like set), the circuit applies the trainable Kraus operators, then the inverse channel, and finally W†. The probability of measuring the all‑zero outcome directly yields the fidelity term. Consequently, the measurement overhead collapses from O(4ⁿ) POVM elements to O(6 n) single‑outcome probabilities, where n is the number of random input states (typically chosen as 6 N for an N‑qubit system).

Optimization proceeds on the Riemannian manifold of CPTP maps. The Euclidean gradient of the cost is projected onto the tangent space, yielding a Riemannian gradient grad C = ∇C − k Sym(k†∇C) for the Kraus case (Sym(A) = (A + A†)/2). Parameter updates use a retraction based on a Cayley transformation, ensuring that the updated Kraus set remains on the Stiefel manifold (k†k = I) and thus respects the CPTP constraints. The same geometric approach applies to the Choi matrix, where the manifold is the cone of positive semidefinite matrices with unit partial trace.

The authors validate CQPT through extensive numerical simulations. First, they reconstruct Haar‑random unitary gates for up to N = 5 qubits. Using only 2ⁿ Kraus operators (far fewer than the full 4ⁿ degrees of freedom) and 6 N random inputs, the algorithm converges to an average infidelity below 1 % for N = 2 and below 3 % for N = 5, with convergence speed decreasing modestly as N grows. Second, they test dephasing channels with both time‑homogeneous and time‑inhomogeneous noise strengths γ. For small γ (near‑unitary regime) the Kraus‑based method recovers the channel with high fidelity; for larger γ the Choi‑based formulation remains stable, achieving comparable performance. Third, depolarizing and amplitude‑damping channels are examined, confirming that the Choi‑based CQPT can handle fully irreversible dynamics without requiring an explicit inverse channel.

Performance comparisons highlight that CQPT reduces measurement complexity from exponential to linear in the number of qubits, while the computational load is dominated by gradient evaluations that scale polynomially (∼2ⁿ × k). The authors implement the optimizer on a workstation equipped with an Intel i9 CPU and an NVIDIA A6000 GPU; typical runs for N = 4 complete within a few minutes, demonstrating practical feasibility.

The paper also discusses limitations. The requirement of an inverse channel is only approximate for non‑unitary processes; the authors address this by linearizing the inverse for weak noise. For strictly irreversible channels, the Choi‑based approach relies on the pseudoinverse, which can be numerically unstable for highly mixed channels. Moreover, the presented scheme is a conceptual algorithm; physical realization of the exact transpose or pseudoinverse operations on current quantum hardware remains an open challenge.

In conclusion, the work proposes a scalable, geometry‑aware QPT method that unifies Kraus and Choi representations under a quantum compilation paradigm. By dramatically lowering measurement overhead and exploiting Riemannian optimization, CQPT offers a viable path toward routine characterization of quantum gates and noisy channels on near‑term quantum processors, with potential extensions to error‑mitigation, device benchmarking, and quantum network verification.