Estimating Non-Stabilizerness Dynamics Without Simulating It

TL;DR

The paper tackles the computational challenge of tracking non-stabilizerness (quantum magic) in large quantum circuits. It introduces Iterative Clifford Circuit Renormalization (ICCR), which renormalizes the initial state to embed magic dynamics into a Clifford-only evolution, paired with a variational MPS description to keep the problem tractable. The method is validated against tensor-network benchmarks and applied to a 1D monitored Clifford circuit, revealing a measurement-induced transition in magic purification at a critical measurement rate. The findings demonstrate that ICCR enables accurate, scalable estimation of magic dynamics in large systems and paves the way for exploring magic-related phase transitions in higher dimensions and more complex circuits.

Abstract

We introduce the Iterative Clifford Circuit Renormalization (ICCR), a novel technique designed to efficiently handle the dynamics of non-stabilizerness (a.k.a. quantum magic) in generic quantum circuits. ICCR iteratively adjusts the starting circuit, transforming it into a Clifford circuit where all elements that can alter the non-stabilizerness, such as measurements or T gates, have been removed. In the process the initial state is renormalized in such a way that the new circuit outputs the same final state as the original one. This approach embeds the complex dynamics of non-stabilizerness in the flow of an effective initial state, enabling its efficient evaluation while avoiding the need for direct and computationally expensive simulation of the original circuit. The initial state renormalization can be computed explicitly using a matrix-product state approximation that can be systematically improved. We implement the ICCR algorithm to evaluate the non-stabilizerness dynamics for systems of size up to N = 1000. We validate our method by comparing it to tensor networks simulations. Finally, we employ the ICCR technique to study a magic purification circuit, where a measurement-induced transition is observed.

Figures (9)

• Figure 1: Example of Clifford circuit with measurements. Measurements of Pauli strings are represented by blue rounded boxes, unitary operators by yellow squared boxes.
• Figure 2: Graphic representation of the T gadget replacement. The blue circle represents a projective measurement of $\hat{Z}$ with positive outcome, as in Fig. \ref{['f:circuit']}.
• Figure 3: Schematic representation of the ICCR algorithm. Projectors on Pauli strings are represented by blue rounded boxes, unitary operators by yellow squared boxes. (a) The original circuit involves a unitary Clifford gate followed by a local projector, in this example corresponding to $\frac{(\hat{\mathds{1}}\pm\hat{Z}_2)}{2}$. (b) The projector is moved through the Clifford gate resulting in a measurement of an extended Pauli string. (c) Single-qubit gates are applied to rotate the new Pauli string into one involving only $\hat{Z}$. (d) The projector is replaced by a cascade of CNOT and single-qubit gates, and the initial state is updated.
• Figure 4: Long-time magic densities (a) $m_2$ and (b) $\nu/N$, as functions of the measurement rate $p$, computed with the ICCR algorithm using different bond dimensions $\chi$. Data for $N=1000$, $T=2000$, averaged over $200$ realizations. All magic densities vanish for $p>p_c\approx 0.16$, whereas the system still features extensive non-stabilizerness below $p_c$.
• Figure 5: Dynamics of the second SRE density $m_2$ for $p>p_c$. Data for $N=1000$ averaged over $200$ realizations. We use $\chi=32$, although other choices of $\chi$ produce qualitatively analogous behavior. The magic relaxes to zero in a finite time with an exponential time dependence.