Optimal Quantum Purity Amplification

TL;DR

The paper solves the long-standing problem of optimal quantum purity amplification (QPA) for general noisy quantum inputs by formulating QPA as a semidefinite program in the Choi representation and exploiting Schur-Weyl symmetry to reduce it to a linear program over irreducible representations. The authors derive the explicit optimal Choi operator, implement it efficiently via a generalized quantum phase estimation framework, and propose SWAPNET for near-term experiments. They demonstrate substantial fidelity gains in digital and analog settings, including simulations of Hamiltonian dynamics and adiabatic state preparation, and validate robustness with experiments on superconducting hardware. The work suggests QPA as a practical subroutine to boost quantum information tasks on NISQ devices and provides a pathway toward platform-agnostic purification with reduced resource overhead. Overall, the results connect deep representation-theoretic structure to a scalable purification protocol with clear experimental routes and performance guarantees.

Abstract

Quantum purity amplification (QPA) provides a novel approach to counteracting the pervasive noise that degrades quantum states. We present the optimal QPA protocol for general quantum systems and global noise, resolving a two-decade open problem. Under strong depolarization, our protocol achieves an exponential reduction in sample complexity over the best-known methods. We provide an efficient implementation of the protocol based on generalized quantum phase estimation. Additionally, we introduce SWAPNET, a sparse and shallow circuit that enables QPA for near-term experiments. Simulations in both digital and analog quantum settings, along with experiments on superconducting quantum processors, confirm the protocol's robustness and practical utility. Our findings suggest that QPA could improve the performance of quantum information processing tasks, particularly in the context of Noisy Intermediate-Scale Quantum (NISQ) devices, where reducing the effect of noise with limited resources is critical.

Figures (26)

• Figure 1: Classical analogy illustrating the QPA protocol $\mathcal{T}$ applied to noisy quantum states. Each of the $n$ copies represents a $d$-dimensional many-body state with low fidelity.
• Figure 2: Illustration of the channel $\mathcal{T}$ for the optimal QPA protocol with $n=5$ and $d>5$. The procedure consists of three steps: Step 1: Schur sampling. The state $\rho^{\otimes n}$ attains a block diagonal form (gray) under the Schur basis. The first step involves projections onto irrep subspaces, each indicated by layers in different colors. Here, only four layers are shown for simplicity. Step 2: Corrections. Corrections are applied to arrange the irreps into the column-ordered form. Step 3: Trace-out. Finally, the last register is output as the state $\rho^\prime = \mathcal{T}(\rho^{\otimes 5})$ while all other registers are discarded.
• Figure 3: Circuit diagram of \ref{['alg:efficient_qpa']}. Step 1: Schur sampling performs weak Schur sampling using GQPE. Step 2: Corrections are done using inverse GQPE. Step 3: Trace-out is performed at the end of the process, the final register with the state $\rho^\prime$ is returned, while all other registers are discarded.
• Figure 4: a Circuit diagram of the SWAPNET algorithm. The ancilla qubit is measured at the end, and based on the outcome $z$, the process either repeats or outputs the purified state $\rho^\prime$. If $z=1$ is never measured, the process terminates after $N_\mathrm{trials}$ iterations, yielding the purified state. b Unrolling the circuit over time, the $z=0(1)$ outcomes of the SWAP tests amount to applying (anti)symmetrizers, represented by white(black) boxes. The outcomes, $\rho^\prime$, are labeled at the returned registers, with the traced-out registers crossed out. c Physical layout for implementation with Rydberg atoms. Resource qubits (including both data qubits and ancillae) are initially stored in the uppermost zone. They are then transferred to the middle zone, where a 3 qubit CSWAP gate is performed using a global pulse. Afterwards, the ancillae qubits are moved to the lowermost zone for readout. d Circuit of the logical CSWAP gate on 3 multiqubit states, achieved by applying transversal CSWAP gates to respective qubits.
• Figure 5: a Base and amplified fidelities for $N_\mathrm{trials} = 1, 2, 3$ as functions of gate error rate $\epsilon$ from Hamiltonian simulation. Inset shows fidelity gain. b Base, amplified, and optimal fidelities for $N_{\mathrm{trials}}=1,2,3$ as functions of gate error rate $\epsilon$ from adiabatic state preparation. Inset shows the top three eigenvalues of the input state. c Base, amplified, and optimal fidelities for $N_{\mathrm{trials}} = 1, 2, 3$ versus depolarization strength $\lambda$ from numerical simulation with global noise only. d, e Base and amplified fidelities for $N_{\mathrm{trials}} = 1, 2, 3$ versus $\lambda$, shown for (d) simulated processor and (e) experimental processor.

Theorems & Definitions (11)

• Definition 1.1: QPA Protocol
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.4: Column-ordered YT
• Proposition S2.1: Invariant Operator
• proof
• Theorem S2.2
• Lemma S2.3
• Lemma S2.4: Equivalence of cost matrixs