The non-stabilizerness cost of quantum state estimation

TL;DR

This work addresses the problem of how much non-stabilizerness (magic) is required to achieve informational completeness in single-setting quantum state estimation using fixed circuits with ancillas. By analyzing stabilizer and non-stabilizer circuits through the stabilizer formalism and gadget constructions, the authors derive quantitative thresholds linking the non-stabilizerness budget (number of $T$ gates) to the accessible information, encapsulated in the dimension $s_{oldsymbol{\mu}}$ of the POVM span. They show that Clifford-only (stabilizer) resources cannot yield IC measurements, establish a lower bound $t \ge \frac{2n}{\log_2 3}$ and a matching sufficient bound $t=2n$ for $t$-doped circuits, and reveal a tight connection between entanglement $p$ in the Heisenberg-evolved measurement states and $s_{oldsymbol{\mu}}$, with IC possible only when $p=n$. These results have practical implications for shadow tomography and quantum machine learning (QELMs) by clarifying the non-stabilizerness costs required to reconstruct observables from fixed measurements, and they propose a concrete framework to design IC measurements within the stabilizer-plus-magic paradigm.

Abstract

We study the non-stabilizer resources required to achieve informational completeness in single-setting quantum state estimation scenarios. We consider fixed-basis projective measurements preceded by quantum circuits acting on $n$-qubit input states, allowing ancillary qubits to increase retrievable information. We prove that when only stabilizer resources are allowed, these strategies are always informationally equivalent to projective measurements in a stabilizer basis, and therefore never informationally complete, regardless of the number of ancillas. We then show that incorporating $T$ gates enlarges the accessible information. Specifically, we prove that at least ${2n}/{\log_2 3}$ such gates are necessary for informational completeness, and that $2n$ suffice. We conjecture that $2n$ gates are indeed both necessary and sufficient. Finally, we unveil a tight connection between entanglement structure and informational power of measurements implemented with $t$-doped Clifford circuits. Our results recast notions of ``magic'' and stabilizerness - typically framed in computational terms - into the setting of quantum metrology.

Figures (6)

• Figure 1: Schematic representation of the types of measurements considered in the paper, with $\Phi$ assumed to be a unitary evolutino $U$.
• Figure 2: Visual representation of the result of \ref{['thm:structural_clifford_povms_plus']}. The generators that determine the effective POVM $\boldsymbol{\mu}$ are elements of $(\mathcal{S}\cap C(\mathcal{Z}'))\setminus \mathcal{Z}'$. More precisely, they are a set of nontrivial representatives for the quotient space $(\mathcal{S}\cap C(\mathcal{Z}')) / (\mathcal{S}\cap \mathcal{Z}')$.
• Figure 3: Serial (left) and parallel (right) doping of an $n$-qubit Clifford circuit. Each layer $\mathcal{C}_0$ is a random Clifford gate. The final measurement is performed in the computational basis.
• Figure 4: Gadget to implement a $T$ gate adding an ancilla initially in $\ket T\equiv T\ket +$. Measuring the second qubit after a CNOT, teleports a $T$ gate on the first qubit when outcome is $\ket 0$, and a $ST$ gate otherwise.
• Figure 5: Example of $t$-doped circuit with $n=m=1$ and $t=2$. Using the gadgets trick, the $T$ gates become CNOTs with target a new "gadget ancilla", whose initial value is $\ket T$ and that is projected onto $\ket0$ at the end of the circuit. The explicit characterisation of this circuit in the stabilizer formalism is given in \ref{['ex:tdoped_2']}.

Theorems & Definitions (25)

• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2
• proof
• Theorem 2
• proof
• Theorem 3
• proof