Magic Measures in Many-body Systems

Besides entanglement, the ‘magic’ of a quantum state is a measure of how useful it is for quantum computation.

Magic means non-stabilizerness, while QC in stabilizer space is proved to have no advantage over classical computation.

Many-body magic measures the resource of states in certain phase.

Page’s theorem: If a quantum state of Hilbert space dimension is in a random pure state,then the average entropy of a subsystem with is
$S{mn}=\sum^{mn}{k=n+1}\frac{1}{k}-\frac{m-1}{2n}$.

Gottesman-Knill theorem: circuits that only consist of:
(i) Preparation on computational basis;
(ii) Clifford gates;
(iii) Measurement on computational basis.
can be perfectly simulated in polynomial time on a probabilistic classical computer.

The Clifford group can be generated solely by using CNOT, Hadamard, and phase gate .

Phase gate : A phase gate represents a rotation along the z-axis on the Bloch sphere by ：

gate is a phase gate with .
gates and gates are also special phase gates.

Reed-Muller code: RM(r,m), encoding -bit massage in bits.

Magic Measures

Relative entropy based measures

Relative entropy: the distance between two states:

$$D(\rho \| \sigma) = \operatorname{Tr} \rho \log \rho -\operatorname{Tr}\rho \log \sigma$$

(1) Min-relative entropy of magic:

$$\mathfrak{D}_{\min }(\rho)=\min _{\sigma \in \mathrm{STAB}} D_{\min }(\rho \| \sigma)$$ $$D_{\min }(\rho \| \sigma) := -\log \operatorname{Tr} \Pi_\rho \sigma$$

: projector to .
For pure states,

$$\mathfrak{D}_{\min} (\psi) = -\log \max_{\phi \in \text{STAB}} |\langle \psi | \phi \rangle|^2$$

(2) Max-relative entropy:

$$\mathfrak{D}_{\max }(\rho)=\min _{\sigma \in \mathrm{STAB}} D_{\max }(\rho \| \sigma)$$ $$D_{\max}(\rho\| \sigma) := \log \min \{\lambda : \rho \leq \lambda \sigma\}$$

(3) Generalized robustness

$$R_g = \min s \geq 0\quad \text{such that} \quad \rho \in (1+s)\text{STAB} - s \mathcal{S}$$

Here denotes the set of all states.
The log-generalized robustness is .

(4) Free robustness

$$R(\rho) = \min s \geq 0 \quad \text{such that} \quad \rho \in (1+s)\text{STAB}-s\text{STAB}$$

The log-free robustness is

Relation to other measures

(1) Stabilizer extent: Equivalent to .

(2) Stabilizer fidelity

(3) Stabilizer rank

(4) Wigner negativity

Too magic to be useful

Consider Pauli MBQC (measurements only on Pauli basis). All magic resource is contained in the initial state.

While standard MBQC with cluster states requires measurements on ‘magical’ basis.

In standard MBQC, if the resource state contains too much entanglement, there will be no quantum speed-up. For Pauli MBQC and magic, there is a similar rule:

Theorem 5 (in paper): Pauli MBQC with any n-qubit resource state that has cannot achieve superpolynomial speedups over BPP machines (classical randomized algorithms) for problems in NP.