Optimal verification of stabilizer states
Ninnat Dangniam, Yun-guang Han, Huangjun Zhu
10 Aug 2020
Department of Physics and Center for Field Theory and Particle Physics, Fudan University
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)
\sigma
State space
Verifying a quantum state
Trusted single-qubit measurement
\(O_A \otimes I_B + I_A\otimes O_B \longrightarrow O_A \otimes O_B\)
classical post-processing
\epsilon
\rho
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
\langle\psi|\sigma|\psi\rangle \\ \le 1-\epsilon
\(XX\)
(relative
phase)
\(ZZ\)
(parity)
|\Phi^+\rangle = |00\rangle +|11\rangle
|\Psi^-\rangle = |01\rangle -|10\rangle
|\Phi^-\rangle = |00\rangle -|11\rangle
|\Psi^+\rangle = |01\rangle +|10\rangle
| +1 | +1 |
| -1 | +1 |
| +1 | -1 |
| -1 | -1 |
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
\delta\approx e^{-N\nu\epsilon}
Chernoff-Stein lemma
\(\sigma=\rho\): Accept with probability 1
(No type I error)
\(\epsilon\)-far: Accept with probability \(\delta\) (type II error)
\Omega|\psi\rangle := \left(\sum_j p_j \Pi_j\right)|\psi\rangle = |\psi\rangle
\(|\psi\rangle\) is a +1 eigenstate of the verification operator
N = \left\lceil
\displaystyle{\frac{\ln\delta^{-1}}{\nu(\Omega)\epsilon}}
\right\rceil
spectral gap \(\nu(\Omega)\)
Pallister et al., PRL 120 170502 (2018)
Zhu & Hayashi, PR Appl. 12, 054047 (2019)
Test projector
Existing results
| Efficient? | Optimal? | |
|---|---|---|
| Bipartite pure states | ||
| GHZ | ||
| Hypergraphs | * | |
| (Phased) Dicke states | ||
| Antisymmetric basis states | ||
| Stabilizer states | ? |
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
GHZ state
- Quantum error correction
- Secret sharing
G=(V,E)
S_j = \bigotimes_{(j,k)\in E} Z_k \otimes X_j
- 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
Graph states
X
Z
Z
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
\Omega:= \sum_j p_j \Pi_j
\Pi_{\mathcal{L}_P} =
\frac{1}{|\mathcal{L}_P|} \sum_{\mathcal{L}_P} s = \prod_{\langle\{s_k\}\rangle = {\mathcal{L}_P}} \displaystyle{\frac{1+s_k}{2}}
\(\mathcal{S}_P\) = stabilizer group generated by all single-qubit Pauli operators that constitute a Pauli measurement
\mathcal{L}_P = \mathcal{S} \cap \mathcal{S}_P
Local subgroup
Example: GHZ state
| XXXX... |
| ZZI... |
| IZZI... |
| IIZZI... |
Measuring ZZ... effectly measures half of the stabilizer group
Canonical test projector
|000\cdots\rangle\langle000\cdots|+|111\cdots\rangle\langle111\cdots|
Verification of stabilizer states
Unique optimal strategy if restricted to stabilizer measurements: measuring all stabilizers with equal probabilities
\nu(\Omega) = \displaystyle{\frac{2^{n-1}}{2^n-1} }\overset{n\to\infty}{\longrightarrow}\displaystyle{\frac{1}{2}}
More general measurements?
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 algorithm
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
s(u)s(v) = i^{2[u,v]} s(v)s(u)
A_u = (M_{\mathcal{S}_u})^T J M_{\mathcal{S}}
J = \begin{pmatrix} 0 & -I_n \\ I_n & 0
\end{pmatrix}
- 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)
| Paulis | Probabilities |
|---|---|
| ZXZXXXX ZZYYXXZ ZYXZXXY XZZYZYX XZXZZZZ XXXXYZY XYYYYYY YZYZZYY YXYXZZZ YYZZYYX YYXYYZZ |
1/6 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 |
The algorithm
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
\tilde{\chi}(G) \le \tilde{\chi}_2(G) \le \chi_{\mathrm{LC}}(G) \le \chi(G)
Equality for all graph states up to seven qubits
Stabilizer verification
By Ninnat Dangniam
Stabilizer verification
TeamNet group seminar, 10 Aug 2020
