Practical Tomography of Multi-Time Processes

Abstract

Characterising multi-time quantum processes is essential for analysing temporally correlated noise and for designing effective control and mitigation strategies. A complete operational description through multi-time process tomography requires an informationally complete set of probes, which necessarily includes non-deterministic intermediate operations. On present-day quantum devices, such operations are commonly implemented using mid-circuit measurements and reset, which are technologically limited and can introduce noise and overhead in terms of ancilla requirement. In this work, we study the minimal ancillary dimension required for complete characterisation of multi-time processes. We show that sequential interactions with a single qubit ancilla can generate an informationally complete family of correlated probes for processes of arbitrary length, without requiring mid-circuit measurements or reset. Our result provides a resource-efficient route for complete multi-time process tomography and establishes that one qubit of coherent ancillary memory suffices for full reconstruction of arbitrary multi-time dynamics.

Figures (4)

• Figure 1: Schematic representing a multi-time process $W$ where the transformations might in general be correlated in time. The probes are depicted by $A_i$ and have associated input and output Hilbert spaces $\mathcal{H}_{A_i^{I}}$ and $\mathcal{H}_{A_i^{O}}$ respectively, where an arbitrary operation can be performed. Multi-time process tomography refers to reconstructing an unknown $W$ through implementing informationally complete set of operations at the probes.
• Figure 2: System--ancilla unitary construction for a single lab operation. The upper wire denotes the system and the lower wire a single-qubit ancilla. If $U^\dagger\left| 0 \right\rangle=\left| a^* \right\rangle$ and $V\left| 0 \right\rangle=\left| \psi \right\rangle$, then the ancilla measurement outcome $\left|0\middle\rangle\!\middle\langle0\right|$ induces the Kraus operator $K_{0,0}=\left|\psi\middle\rangle\!\middle\langle a^*\right|$, whose Choi operator is $\left|K_{0,0}\middle\rangle\!\middle\langle K_{0,0}\right|=\left|a\middle\rangle\!\middle\langle a\right|^{T}\otimes\left|\psi\middle\rangle\!\middle\langle\psi\right|$. Varying $\left| a \right\rangle$ and $\left| \psi \right\rangle$ generates a family of measure-and-prepare operations that spans the full operator space.
• Figure 3: (a) Schematic depiction of measure and rotation resulting in the operations $|a\rangle\langle a|^T\otimes |\psi\rangle\langle\psi|$, which forms an informationally complete set of operations, however, difficult to implement in present devices. (b) Alternative implementation of measure and prepare operation through an ancilla and swap operation. (c) Measure and prepare in all the intermediate probes can be implemented if there are no restrictions on the number and the dimension of ancilla. However, it comes with a significant resource cost in addition to the implementation issue pertaining to applying SWAP with non-neighbouring ancillas.
• Figure 4: (a) A depiction of correlated operation across two intermediate labs $T_a$ to characterise the process $W$ (b) An implementation of the correlated instrument $T_a$ through continuous interaction with a single ancilla. The resulting operation has a restricted MPO structure.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Lemma 1