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Stabilizer Ranks, Barnes Wall Lattices and Magic Monotones

Amolak Ratan Kalra, Pulkit Sinha

TL;DR

This paper advances the quantitative understanding of stabilizer ranks in quantum circuit simulation by linking stabilizer fidelity to stabilizer rank through a lattice-based Gram determinant bound, and by introducing the Barnes Wall norm $\mathcal N$ and its approximate variant $\mathcal N_{\delta}$ as Clifford-invariant magic monotones. It proves a first explicit lower bound on stabilizer fidelity in terms of stabilizer rank, yielding a linear-by-log lower bound $\chi( |\psi\rangle )=\Omega(n/\log n)$ when the overlap with $|H\rangle^{\otimes n}$ is exponentially large, and develops Fidelity Amplification to trade rank against approximation error and to compose stabilizer decompositions across tensor powers. The work also establishes a tight connection between lattice-approximation quality and stabilizer ranks, and provides an elementary density result showing that product states with maximal stabilizer rank are open and dense. Collectively, these results unify lattice-theoretic methods with stabilizer-rank analyses, yielding new lower bounds, construction techniques, and a pathway to understanding the complexity of approximate stabilizer decompositions for tensor powers of magic states.

Abstract

In 2024, Kliuchnikov and Schönnenbeck showed a connection between the Barnes Wall lattices, stabilizer states and Clifford operations. In this work, we study their results and relate them to the problem of lower bounding stabilizer ranks. We show the first quantitative lower bound on stabilizer fidelity as a function of stabilizer ranks, which reproduces the linear-by-log lower bound for $χ_δ({|{H}\rangle^{ \otimes n}})$, i.e, on the approximate stabilizer rank of $|H\rangle^{\otimes n}$. In fact, we show that the lower bound holds even when the fidelity between the approximation and ${|H\rangle}^{\otimes n}$ is exponentially small, which is currently the best lower bound in this regime. Next, we define a new magic monotone for pure states, the Barnes Wall norm, and its corresponding approximate variant. We upper bound these monotones by the $CS$-count of state preparation, and also by the stabilizer ranks. In particular, the upper bound given by the $CS$-count is tight, in the sense that we exhibit states that achieve the bound. Apart from these results, we give a Fidelity Amplification algorithm, which provides a trade-off between approximation error and the stabilizer rank. As a corollary, it gives us a way to compose approximate stabilizer decompositions into approximate decompositions of their tensor products. Finally, we provide an alternate, elementary proof of the existence and density of product states with maximal stabilizer ranks, which was first proven by Lovitz and Steffan (2022), where they used results from algebraic geometry.

Stabilizer Ranks, Barnes Wall Lattices and Magic Monotones

TL;DR

This paper advances the quantitative understanding of stabilizer ranks in quantum circuit simulation by linking stabilizer fidelity to stabilizer rank through a lattice-based Gram determinant bound, and by introducing the Barnes Wall norm and its approximate variant as Clifford-invariant magic monotones. It proves a first explicit lower bound on stabilizer fidelity in terms of stabilizer rank, yielding a linear-by-log lower bound when the overlap with is exponentially large, and develops Fidelity Amplification to trade rank against approximation error and to compose stabilizer decompositions across tensor powers. The work also establishes a tight connection between lattice-approximation quality and stabilizer ranks, and provides an elementary density result showing that product states with maximal stabilizer rank are open and dense. Collectively, these results unify lattice-theoretic methods with stabilizer-rank analyses, yielding new lower bounds, construction techniques, and a pathway to understanding the complexity of approximate stabilizer decompositions for tensor powers of magic states.

Abstract

In 2024, Kliuchnikov and Schönnenbeck showed a connection between the Barnes Wall lattices, stabilizer states and Clifford operations. In this work, we study their results and relate them to the problem of lower bounding stabilizer ranks. We show the first quantitative lower bound on stabilizer fidelity as a function of stabilizer ranks, which reproduces the linear-by-log lower bound for , i.e, on the approximate stabilizer rank of . In fact, we show that the lower bound holds even when the fidelity between the approximation and is exponentially small, which is currently the best lower bound in this regime. Next, we define a new magic monotone for pure states, the Barnes Wall norm, and its corresponding approximate variant. We upper bound these monotones by the -count of state preparation, and also by the stabilizer ranks. In particular, the upper bound given by the -count is tight, in the sense that we exhibit states that achieve the bound. Apart from these results, we give a Fidelity Amplification algorithm, which provides a trade-off between approximation error and the stabilizer rank. As a corollary, it gives us a way to compose approximate stabilizer decompositions into approximate decompositions of their tensor products. Finally, we provide an alternate, elementary proof of the existence and density of product states with maximal stabilizer ranks, which was first proven by Lovitz and Steffan (2022), where they used results from algebraic geometry.

Paper Structure

This paper contains 16 sections, 29 theorems, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of the Main Results
  3. Preliminaries
  4. Arithmetic for $\mathbb{Z}[i]$
  5. Background on Barnes Wall Lattices
  6. Linear Lower Bound on Stabilizer Rank
  7. Magic Monotones and Barnes Wall Lattices
  8. Properties of $\mathcal{N}$ and $\mathcal{N}_\delta$
  9. Connection Between $\mathcal{N}_{\delta}$ and Stabilizer Ranks
  10. Relating $\mathcal{N}$ and $CS$-count for exact state preparation
  11. Fidelity Amplification
  12. Relation to Lattice Approximations
  13. Exponential Exact Stabilizer Rank for a Product State
  14. Density and Openness of Maximum Rank States
  15. Directions for Future work
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathinner{|{\phi}\rangle}$ be a pure state such that $\mathinner{|{\phi}\rangle}=\sum_{j=1}^k c_i\mathinner{|{s_j}\rangle}$, where $\mathinner{|{s_j}\rangle}$'s are all linearly independent stabilizer states, $c_j\in \mathbb C$. Then, we have that $\left\| \mathbf c \right\|_1\leq \sqrt e\cdot

Figures (1)

  • Figure 1: $CS$-gate injection using the magic state $\mathinner{|{CS}\rangle}$.

Theorems & Definitions (65)

  • Theorem 1: Upper bound on stabilizer extent
  • Theorem 2: Lower bound on stabilizer fidelity
  • Corollary 3: Linear-over-log lower bound
  • Corollary 4
  • Definition
  • Theorem 5: Magic monotone properties
  • Theorem 6
  • Theorem 7
  • Theorem 8: Fidelity amplification for $\mathinner{|{H}\rangle}^{\otimes n}$
  • Corollary 9
  • ...and 55 more