Identitifying a quantum state

Trusted multi-qubit measurement

• Quantum state tomography fully identifies an unknown quantum state
• Sample-optimal tomography requires an exponential amount of resource (Haah et al., 2017), entangled measurements (serial or parallel)

i.i.d. quantum states

$\sigma^{\otimes N}$ (possibly mixed)

Verifying a quantum state

Trusted single-qubit measurement

i.i.d. quantum states

$\sigma^{\otimes N}$ (possibly mixed)

• Quantum state verification aims to certify that $\sigma$ is close to some target pure state $\rho$ using only local projective measurements

State space

$XX$

(relative

phase)

$ZZ$

(parity)

For general pure states (linear combinations of the four Bell states) and mixed states, there will be a failure probability to detect the target state 

Example: Distinguishing Bell states

($|\Phi^+\rangle\langle\Phi^+|$ is nonlocal)

Measure $\{\Pi_j, I-\Pi_j\}$ with probability $p_j$

Obtain

$\Pi_j$

Obtain

$I-\Pi_j$

Reject

The hypothesis testing scenario

Chernoff-Stein lemma

$\sigma=\rho$: Accept with probability 1

(No type I error)

$\epsilon$-far: Accept with probability $\delta$ (type II error)

$|\psi\rangle$ is a +1 eigenstate of the verification operator

spectral gap $\nu(\Omega)$

Pallister et al., PRL 120 170502 (2018)

Zhu & Hayashi, PR Appl. 12, 054047 (2019)

Test projector

Existing results

Pallister et al., PRL 120 170502 (2018)

Our work

Main results for this talk

• There is an efficiency bound for verification of stabilizer states that does not depend on the number of qubits or the specific stabilizer states
• The bound can be saturated, at least up to seven qubits, by explicitly constructing an optimal measurement scheme

Stabilizer and graph states

Stabilizer states

Efficiently describable states

Cluster states

• Resource states for MBQC

• Every stabilizer state is LU-equivalent to some graph state

• Stabilizer state = stabilized by a commuting subgroup of the $n$-qubit Pauli group

$\mathcal{P}_n = \{\pm 1,\pm i\}\cdot\{I,X,Y,Z \}^{\otimes n}$

       that does not contain $-I$, called a stabilizer group $\mathcal{S}$

• Completely speficified by $n$ independent generators of the group

Every convex combination of stabilizers in $\mathcal{S}$ of a target stabilizer state $|\psi\rangle$ has $|\psi\rangle$ as a +1 eigenstate, but not all of them are measurable locally

Characterizing test projectors

$\mathcal{S}_P$ = stabilizer group generated by all single-qubit Pauli operators that constitute a Pauli measurement

Local subgroup

Example: GHZ state

Measuring ZZ... effectly measures half of the stabilizer group

Canonical test projector

Verification of stabilizer states

Unique optimal strategy if restricted to stabilizer measurements: measuring all stabilizers with equal probabilities 

Pallister, Linden and Montanaro, PRL 120 170502 (2018)

Wang and Hayashi, arXiv:1901.09467 [quant-ph]

(2/3 when $n=2$)

• Increasing the dimension of one half of a maximally entangled bipartite system cannot increase $\nu(\Omega) = 2/3$ 
• But every stabilizer state is maximally entangled w.r.t. any $(\mathcal{C}^2, (\mathcal{C}^2)^{\otimes n})$-bipartition

Spectral gap is upper bounded by 2/3

• Increasing the dimension of one half of a maximally entangled bipartite system cannot increase $\nu(\Omega) = 2/3$ 
• But every stabilizer state is maximally entangled w.r.t. any $(\mathcal{C}^2, (\mathcal{C}^2)^{\otimes n})$-bipartition

Spectral gap is upper bounded by 2/3

The bound can be saturated for all graph states up to seven qubits by explicitly constructing the optimal measurement scheme

• All computation is done in the symplectic vector formalism

• Construct the local stabilizer projector.
• Optimize the spectral gap ($l_{\infty}$-norm minimization - a linear program hence efficient)
• Do this for all 45 LU-equivalent classes of graph states (LC-equivalent to stabilizer states)

Hein, Eisert and Briegel, PRL 69 062311 (2004)

Other results

• Necessary condition for optimal verification by Pauli measurements $p_x=p_y=p_z=1/3$
• Optimal XZ strategies ($\nu = 1/2$), optimal strategy for odd ring states (simplest non 2-colorable graph)
• Admissible test projector = canonical test projectors that do not contain any other test projector, admissibility of XZ measurements $\iff$ maximal independent set
• Minimal number of settings for verification

Equality for all graph states up to seven qubits