Relating Non-stabilizerness to Other Quantum Resources

Tanausú Hernández-Yanes

Chaos i Informacja Kwantowa

2025-10-27

Non-stabilizerness and Quantum Computing

A state can achieve quantum advantage only if it contains a sufficient amount of non-stabilizerness, measured by, for instance, the stabilizer Rényi entropy \(\mathcal{M}_q\).

Quantum error correction
Fault-tolerant quantum computing

Non-stabilizerness in Particle Physics

Shalma Wegsman, Particle Physicists Detect ‘Magic’ at the Large Hadron Collider, Quanta Magazine (2025)

The Stabilizer Polytope

For a single qubit, the polytope is defined by the convex mixture of its six Pauli eigenstates

\[\text{conv} \left\{\frac{1+\vec{a}\cdot\vec{\sigma}}{2}\right\}_{\vec{a}\in{\pm \vec{x}, \pm \vec{y}, \pm \vec{z}}} \]

The Stabilizer Polytope

\[ \rho = \frac{1}{2}(1 + \vec{r}\cdot\vec{\sigma}) \rightarrow |r_x| + |r_y| + |r_z| \le 1 \]

\[ \mathcal{NS}(\rho) = \min_{\sigma\in\text{STAB}} \frac{1}{2}|| \rho - \sigma||_1 \]

Stabilizer States

Construction analogous to classical linear codes
Eigenstates of the tensor product of local Pauli operators with eigenvalue \(+1\) ( \(\hat{P} \ket{\psi} = \ket{\psi}\), \(\hat{P}\) stabilizes \(\ket{\psi}\) )
Highly entangled states can also be stabilizer states, see \(\hat{X}_1\hat{X}_2\) and \(\hat{Z}_1\hat{Z}_2\) acting on \(\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})\)

Why Are Stabilizer Resources Free?

Many states are easier to describe through the operators that stabilize them
Gottesman-Knill theorem: stabilizer states can be simulated classically in polynomial time
One extra operation needed for universal set for quantum computation

Stabilizer formalism

The Pauli group, which consists of all Pauli matrices, together with multiplicative factors \(\pm 1, \pm i\) \(G_1 \equiv \{\pm \hat{\mathcal{I}}, \pm i\hat{\mathcal{I}}, \pm \hat{X}, \pm i\hat{X}, \pm \hat{Y},\pm i\hat{Y}, \pm \hat{Z}, \pm i\hat{Z}\}\) is closed under multiplication¹.
For \(N\) qubits, \(G_N\) is the \(N\)-fold tensor product of the group.
For a subgroup \(S\in G_N\), the vector space \(V_S\) is stabilized by \(S\) if every element of \(V_S\) is stable under action of any element in \(S\).

Stabilizer formalism

Group description is more compact through generators \(\hat{X}, \hat{Z}\). Any non-trivial subgroup \(S\) requires

Elements of \(S\) commute (Abelian)
\(\{-\hat{\mathcal{I}}\}\notin S\)

Clifford unitary operations map Pauli string to Pauli string (permutations)

Stabilizer formalism

Stabilizer operations are compositions of:

Clifford unitaries
Measurements in the computational basis

Relative entropy is a well defined monotone for any resource

\[ r_\mathcal{M} (\rho) = \min_{\sigma\in \text{STAB}} S(\rho || \sigma) \]

Stabilizer Rényi Entropy (SRE)

\[\begin{equation} \mathcal{M}_q(|\psi\rangle)=\frac{1}{1-q}\log_2\!\left(\frac{1}{2^N}\sum_{\hat{P}\in\mathcal{P}_N}\!\!\langle \psi|\hat{P}| \psi \rangle^{2q}\right) \end{equation}\]

where \(\hat{P}\) is a Pauli string, a tensor product of local Pauli operators and the identity.

Monotone
Non-negative

Clifford invariant
Additive on product states

Non-stabilizerness in relation to other quantum resources

Non-stabilizerness and Entanglement

Study of divide from a computation perspective
Separation based on stabilizer nullity \(\nu\) and entanglement entropy \(S_1\)

Non-stabilizerness and Entanglement

Entanglement estimation \(S_1(\psi_A)\) efficient in ED, inefficient beyond logarithmic entanglement in MD
Entanglement distillation (distills \(M_+\) Bell pairs from \(\psi\) using LOCC) in poly. time in ED, impossible in MD
Entanglement dillution (prepares \(\psi\) across the bipartition \(A|B\) using LOCC, \(M_-\) Bell pairs) is optimal in ED, suboptimal in MD

Non-stabilizerness and Fermionic Non-Gaussianity

Fermionic Gaussian states and unitaries constructed from Majorana fermion operators
Analogous construction to stabilizer states

Non-stabilizerness and Fermionic Non-Gaussianity

Wick’s theorem tells us they are simulatable in polynomial time
Non-Gaussianity can be measured by fermionic anti-flatness \(\mathcal{F}_k\)

Universal Operation Set for Permutationally Symmetric States

Bosonic states can be universally manipulated with operations that mix Clifford and non-Clifford operations
For permutationally symmetric states, this universal set can be described with rotations and one-axis twisting only¹

Non-stabilizerness in Relation to Other Quantum Phenomena

Stabilizer states are completely non-contextual¹
Quantum chaos is exponentially expensive for classical simulation based on stabilizer formalism²

Non-stabilizerness in Relation to Other Quantum Phenomena

Non-stabilizerness can be probed using Bell inequalities¹
Thermal effects can produce distillable non-stabilizerness²

Non-stabilizerness in Quantum-Enhanced Metrological Protocols

The Team

Motivation

Non-stabilizerness is an interesting new quantum resource that is receiving a lot of attention
Implications on many-body systems relevant for quantum sensing
One-axis twisting is a clean metrological protocol available in many different platforms

The computational complexity of the SRE is exponential, as we need to obtain \(4^N\) expectation values.

Said complexity is reduced in the permutation invariant sector to polynomial¹, \(\mathcal{O}(N^3)\).

Can we do better?

Approximated SRE for Permutation Invariant Sector and \(N\to\infty\)

Approximated SRE

\[\begin{equation} \lim_{N \to \infty} \mathcal{M}_q \simeq \frac{1}{1-q}\log_2\left( \frac{1}{2} \sum_{\sigma\in \{X,Y,Z\}} \sum_{n=1}^2 \sum_{m=1}^2 \left(c_{n, m}^{(\sigma)}\right)^{2q} \right) ,\end{equation}\]

with coefficients

\(c^{(\sigma)}_{1,m} = |\bra{\sigma}\ket{\psi}|^2 + (-1)^m |\bra{-\sigma}\ket{\psi}|^2\), \(c^{(\sigma)}_{2,m} = \sqrt{(-1)^m}(\bra{\psi}\ket{-\sigma}\bra{\sigma}\ket{\psi} +(-1)^m \bra{\psi}\ket{\sigma}\bra{-\sigma}\ket{\psi})\).

Representation of \(\hat{P}\) in SU(2)

A natural choice is to use the angular momentum eigenstates \(\ket{J = N/2, m}\).

We choose instead the overcomplete basis of spin coherent states

\[\begin{align} \ket{\theta, \phi} =& \bigotimes_{j=1}^{2J} \left( \cos({\theta}/{2})\ket{0} + e^{-i\phi}\sin({\theta}/{2})\ket{1} \right) \\ =& \sum_{m=-J}^{J} \sqrt{2J \choose J+m} \cos^{J-m} \frac{\theta}{2} \left(\sin\frac{\theta}{2} e^{-i\phi}\right)^{J+m}\ket{J, m} \end{align}\]

Representation of \(\hat{P}\) in SU(2)

\[\begin{multline} \hat{\mathbb{I}}_{J}\hat{P}\hat{\mathbb{I}}_{J} = \left( \frac{2J+1}{4\pi} \right)^2 \int \int d\vb*{\Omega} d\vb*{\Omega'} \ket{\theta, \phi} \bra{\theta', \phi'} \\ \class{fragment grow highlight-blue}{{} \cdot \bra{\theta, \phi} \hat{P} \ket{\theta', \phi'}} \end{multline}\]

Comparing Stabilizer Rényi Entropy Scalings

\(\ket{\theta = \acos(3^{-1/2}), \phi = \pi / 4 }\) (black), best squeezed state for TACT (purple), and kitten state with \(n=8\) (orange).

One-Axis Twisting (OAT)

\[\begin{equation} \ket{\psi(t)} = \exp\left\{-i \frac{\chi t}{4 \hbar}\hat{Z}^2 \right\} \ket{X},\quad \hat{Z} \equiv \sum_j \hat{Z}_j \end{equation}\]

\[ \xi^2 = N \frac{\min_{\vb*{n}} (\Delta \hat{\sigma}_{\perp, \vb*{n}}^2 )}{\ev*{\hat{X}}^2} \gtrsim N^{-2/3} \]

SRE at Squeezing Time Scale

\(t \lesssim t_\mathrm{best}\)

\(N=100\)

SRE Scaling of Squeezed States

SRE and Many-body Bell Correlations

\[\begin{equation} Q = \log_2(2^N {\mathcal E})<0,\quad {\mathcal E}= \left|\frac{1}{N!}\ev{\left(\frac{\hat Y +i\hat Z}{2}\right)^N}\right|^2 \end{equation}\]

\(Q \leq 0\) for separable states
\(Q \geq N - k\) detects non-\(k\)-separability
\(Q = N - 2\) for the GHZ state along x-axis

\(Q\) Scaling for Squeezed States

\[ Q(t = t_\mathrm{best}) \approx N - aN^{1/3} \]

At \(t_\mathrm{best}\), SRE scales with \(\log_2(N)\) while \(Q\) scales sub-linearly with \(N\).

What about other metrologically useful states?

SRE of Kitten States

\(\chi t \leq \frac{\pi}{2}\)

\(N=100\)

\(Q\) and SRE Scaling of Kitten States

SRE is fixed, but grows with \(n\), while \(Q \simeq N-2\log_2(n)\). Anti-correlation of SRE and \(Q\) for kitten states.

Conclusions

Approximated stabilizer Rényi entropy (SRE) for permutation symmetric sector at large \(N\) only requires six state projections
Spin squeezing generation implies an SRE, non-stabilizerness, increase
Fixing the squeezing level also fixes the SRE
The SRE of kitten states is fixed for large \(N\) but grows with the kitten size \(n\), while larger kittens imply smaller Bell correlations \(Q\)

Thank You

Complementary Results

Two-axis Counter-twisting (TACT)

\[\begin{equation} \ket{\psi(t)} = \exp\left\{-i \frac{\chi t}{4 \hbar}(\hat{Z}\hat{Y} + \hat{Y}\hat{Z}) \right\} \ket{X} \end{equation}\]

OAT, \(N=100\), \(\xi^2 \gtrsim N^{-2/3}\).

TACT, \(N=100\), \(\xi^2 \gtrsim N^{-1}\).

SRE of Squeezed States under TACT

\(N=100\)

Same conclusions as OAT.

SRE Scaling of Squeezed States under TACT

Same conclusions as OAT.

Dicke States with Zero Magnetization

\[\begin{equation} \lim_{N\to\infty} \mathcal{M}_q(\ket{J = N/2, 0}) \simeq \frac{q}{q-1}\log_2\left(\frac{\pi N}{8}\right) - \frac{1}{q-1} \end{equation}\]

SRE scaling of best squeezed states under OAT and TACT, and \(\ket{J = N/2, 0}\).

\[ Q \approx N - \log_2 N + \log_2 \frac{2}{\pi} \]

Generalized GHZ States

\[\begin{equation} \frac{1}{\sqrt{K}}\left( \ket{\theta=0, \phi=0} + \ket{\theta = 2\epsilon, \phi=0} \right) \end{equation}\]

Extra Slides

Matrix elements of \(\hat{P}\)

\[\begin{equation} \bra{\theta_i, \phi_i}\hat{P}\ket{\theta_j, \phi_j} = (\alpha_{i,j})^{N_X} (\beta_{i,j})^{N_Y} (\gamma_{i,j})^{N_Z} (\kappa_{i,j})^{N_I} ,\end{equation}\] where \(N_\sigma\) denotes the number of distinct Pauli operators \(\hat{\sigma}\) composing the Pauli string.

Implicit constraints:

\(0 \le x \le 1,\quad \forall x \in \{|\alpha_{i,j}|^2, |\beta_{i,j}|^2, |\gamma_{i,j}|^2, |\kappa_{i,j}|^2\}\),

\(|\alpha_{i,j}|^2 + |\beta_{i,j}|^2 + |\gamma_{i,j}|^2 + |\kappa_{i,j}|^2 = 2\).

Feasible constraints on \(\hat{P}\)

\(x_1, x_2, x_3, x_4 \in \mathcal{P}\{|\alpha_{i,j}|^2, |\beta_{i,j}|^2, |\gamma_{i,j}|^2, |\kappa_{i,j}|^2\}\)

No constraints: complicated integral with no advantage over literature results.
\(x_1 = x_2 = 1\): we only consider matrix elements independent of the system size \(N\).
\(x_1 = 1\): solvable through beta and gamma functions that yield expression (preliminary, unchecked) which only depends on \(\ket{-Z} = \otimes_{j=1}^N \ket{0}\).

Approximated \(\ev*{\hat{P}}\)

\[\begin{equation} \ev*{\hat{P}} \simeq \sum_{\sigma \in \{X,Y,Z\}} \Bigg( c_{1,N_\sigma}^{(\sigma)} \bra{\sigma}\hat{P}\ket{\sigma} + \frac{c_{2,N_{\overline{ \sigma}}}^{(\sigma)}}{\sqrt{(-1)^{N_{\overline{ \sigma}}}}} \bra{\sigma}\hat{P}\ket{-\sigma} \Bigg) \end{equation}\]

Approximated SRE

After some combinatorial calculations, we obtain

Husimi functions for OAT and \(t < \pi / 2\)

\(Q\) scaling for fixed squeezing level

Coherence and SRE under OAT

\(N=100\)

Dicke states scaling

Corrections to the Approximated SRE

Example state: Generalized GHZ state

\[ \ket{\theta = 0, \phi = 0} + \ket{\theta = \pi/3, \phi = 0} \]

Pauli spectra of generalized GHZ state

\(N=50\)

Example state: Rotated coherent state

\[ \ket{\theta = \acos(3^{-1/2}), \phi = \pi/4} \]

Pauli Spectra of rotated coherent state

\(N=50\)

Example state: Dicke state with magnetization \(N/4\)

\[\ket{J=N/2, m = N/4}\]

Pauli spectra of Dicke state with magnetization \(N/4\)

\(N=50\)

Magic Resources for Fault-tolerant Quantum Computing

Concatenation could outperform surface code
No universal set of gates with transversal gates only
Non-transversal gates can be generated with non-stabilizer states and post-selection¹
Surprisingly cheap compared to other requirements²