Optimal Qubit Purification and Unitary Schur Sampling via Random SWAP Tests

TL;DR

This work shows that a simple random SWAP-test protocol suffices to implement Schur sampling on permutationally invariant qubit states, matching the Schur transform’s fidelity with exponentially small error in the number of tests. Leveraging SU(2) symmetry and Schur-Weyl duality, the authors prove reversibility within PI sectors and derive a sharp threshold around $T^* \approx 2n \ln n$, with a concrete bound $D \le n e^{-T/(2n)}$. The approach yields optimal qubit purification fidelity comparable to Schur-based methods, requiring only $O(n)$ SWAP tests and offering variants including compression and a fully coherent version. These results position random SWAP tests as a powerful, lossless subroutine for PI information tasks such as state tomography and metrology, while maintaining minimal experimental overhead.

Abstract

The goal of qubit purification is to combine multiple noisy copies of an unknown pure quantum state to obtain one or more copies that are closer to the pure state. We show that a simple protocol based solely on random SWAP tests achieves the same fidelity as the Schur transform, which is optimal. This protocol relies only on elementary two-qubit SWAP tests, which project a pair of qubits onto the singlet or triplet subspaces, to identify and isolate singlet pairs, and then proceeds with the remaining qubits. For a system of $n$ qubits, we show that after approximately $T \approx n \ln n$ random SWAP tests, a sharp transition occurs: the probability of detecting any new singlet decreases exponentially with $T$. Similarly, the fidelity of each remaining qubit approaches the optimal value given by the Schur transform, up to an error that is exponentially small in $T$. More broadly, this protocol achieves what is known as weak Schur sampling and unitary Schur sampling with error $ε$, after only $2n \ln(n ε^{-1})$ SWAP tests. That is, it provides a lossless method for extracting any information invariant under permutations of qubits, making it a powerful subroutine for tasks such as quantum state tomography and metrology.

Figures (5)

• Figure 1: Qubit Purification and Unitary Schur Sampling via Random SWAP Tests. This protocol uses random SWAP tests to detect and isolate pairs of qubits in the singlet state $(|01\rangle - |10\rangle)/\sqrt{2}$. In the figure above, the partitions $B_k$, for $k = 1, \dots$, correspond to the detected and separated singlets. The partitions $A_k$, for $k = 1, \dots$, contain the remaining qubits. At each step $k$, we randomly select a pair of qubits from partition $A_k$, perform a SWAP test on them, and if the outcome projects onto the singlet state, we move that pair to partition $B_k$ and proceed to step $k+1$. Otherwise, if the SWAP test does not detect a singlet, we randomly choose another pair from $A_k$ and repeat the process. We prove that after $T \ge 2n\ln (n \epsilon^{-1})$ SWAP tests, with probability at least $1-\epsilon$, all singlets in the system have been detected. Furthermore, for the initial state $\rho^{\otimes n}$ with $\rho = (1-p)|\psi\rangle\langle\psi| + p\mathbb{I}/2$, where $|\psi\rangle$ is unknown, any qubit in $A_k$ will have the highest achievable fidelity given in Eq. (\ref{['opt-Fid']}), up to an additional error of at most $\epsilon$.
• Figure 2: The fidelity of the output qubit of this purification protocol, with the desired ideal pure state, after $T$ SWAP tests. Initially the system contains $n=100$ copies of $\rho=(1-p)|\psi\rangle\langle\psi| + p\mathbb{I}/2$. As the number of SWAP tests grows, the fidelity of the output state converges to the optimal fidelity given by Eq.(\ref{['opt-Fid']}). The vertical dotted line corresponds to $n \ln n$, which roughly indicates the order of the number of required SWAP tests. See Sec.\ref{['Sec:App:Pur']} for further details.
• Figure 3: The quantum circuit for implementing the SWAP test. In the fully coherent version of the protocol discussed in Sec.\ref{['coherent']} the final measurement is omitted.
• Figure 4: Eq. (\ref{['TD']}) shows that the error in this protocol, as quantified by the trace distance with the output of the Schur transform, for input states restricted to a sector with angular momentum $j$, is exactly equal to the probability that not all of the existing $\frac{n}{2} - j$ singlets in the system have been detected. Top plot shows this probability as a function of $T$, the number of SWAP tests, for a system with $n=1000$ qubits. Bottom plot shows $T_\ast(j)$, the expected number of SWAP tests required to detect all ${n}/{2} -j$ singlets, again for the same number of $n=1000$ qubits.
• Figure 5: The nodes represent the state of register $|j'\rangle$, or, equivalently, the number of detected singlets, namely $k=n/2-j'$ singlets. Initially, the register is at $j'=n/2$, which means we have not detected any singlet yet. If the input state $\tau_j$ is restricted to the sector with angular momentum $j$, then in the limit $T\gg 1$ SWAP tests, the register goes to state $|j\rangle$ with probability 1, i.e., we detect all the existing $n/2-j$ singlets. The process can be understood as a Markov chain: When the register is in state $j'$ the probability that a SWAP test detect a singlet is $e_{j'}(j)$ given in Eq.(\ref{['prob']}), which means with probability $1-e_{j'}(j)$ the register remains in the same state.

Theorems & Definitions (3)

• Lemma 1
• proof
• Lemma 2