This presentation would have been given at the 2020 APS March Meeting. Title: Variational Quantum Fidelity Estimation Speaker: Marco Cerezo Abstract: We present an efficient, near-term algorithm for estimating the well-known fidelity, which quantifies the closeness of quantum states. Our algorithm is an important tool for verifying and characterizing states on a quantum computer. This work is timely given the industrial rise of quantum computing. Prior to our work, there was no efficient algorithm to estimate the fidelity that could be refined to arbitrary tightness. We solve this outstanding problem by introducing novel bounds on the fidelity that can be estimated with hybrid quantum-classical computation. We show that our approach can detect quantum phase transitions and cannot be classical simulated efficiently. Follow Marco on Twitter: https://twitter.com/MvsCerezo

1. Los Alamos National Laboratory
Variational Quantum Fidelity Estimation
APS March Meeting
March 2020 @ Internet
Marco Cerezo
T-4 / Center for Nonlinear Studies (CNLS)
LA-UR- 19-25585
Managed by Triad National Security, LLC for the U.S. Department of Energy's NNSA
In collaboration with A. Poremba, L. Cincio, and P. Coles
arXiv:1906.09253 (accepted in Quantum)
cerezo@lanl.gov arXiv:1906.09253 (accepted in Quantum)

2. Los Alamos National Laboratory
3/6/2020 | 2
Outline of Main Results
Quantum Information Theory
• Novel upper and lower bounds for the Quantum Fidelity 𝐹 ,  .
• “Truncated Fidelity”, projecting a quantum state  into its m-largest eigenvalues.
Variational Hybrid Quantum-Classical Algorithm
• Variational Quantum Fidelity Estimation Algorithm.
• Numerical implementations for low Rank states.
Complexity Analysis
• The problem of “Low Rank Fidelity Estimation” is classically hard.
cerezo@lanl.gov arXiv:1906.09253

3. Los Alamos National Laboratory
3/6/2020 | 3
Motivation: Near-term quantum devices as (impure)
quantum state preparation factories.
cerezo@lanl.gov arXiv:1906.09253
• Intentionally: thermal states
• Unintentionally: quantum noise, e.g., T1, and T2 processes
Quantum Fidelity as a measure for verification and
characterization
𝐹 ,  = Tr   =   1
 Classically hard to compute (exponentially hard!)
 No quantum algorithm to exactly compute 𝐹 , 

4. Los Alamos National Laboratory
3/6/2020 | 4
• 𝑟𝑖, |𝑟𝑖 : eigenvalues and eigenvectors . Define Π 𝑚

as the projector onto the subspace
of the eigenvectors of  with 𝑚-largest eigenvalues. Define the sub-normalized states:
 𝑚 = Π 𝑚

Π 𝑚

= 𝑖=1
𝑚
𝑟𝑖|𝑟𝑖 𝑟𝑖|,  𝑚

= Π 𝑚

Π 𝑚

(𝑟𝑖≥ 𝑟𝑖+1)
Truncated Fidelity 𝐹  𝑚,  = Tr  𝑚  1
Truncated Generalized Fidelity1 𝐹∗  𝑚,  = Tr  𝑚  1
+ (1 − Tr  𝑚)(1 − Tr  𝑚

)
Truncated Fidelity Bounds (collection of bounds with 𝑚, Fidelity Spectrum):
𝐹  𝑚,  ≤ 𝐹 ,  ≤ 𝐹∗  𝑚, 
 Bounds get monotonically tighter with 𝑚:
𝐹  𝑚,  ≤ 𝐹  𝑚+1,  𝐹∗  𝑚+1,  ≤ 𝐹∗  𝑚, 
[1] M. Tomamichel, “Quantum Information Processing with Finite Resources: Mathematical Foundations”, Vol. 5 (Springer,2015).
Bounding the fidelity: Truncated Fidelity Bounds
cerezo@lanl.gov arXiv:1906.09253

5. Los Alamos National Laboratory
3/6/2020 | 5
Truncated Fidelity 𝐹  𝑚,  = Tr  𝑚  1
Truncated Generalized Fidelity 𝐹∗  𝑚,  = Tr  𝑚  1
+ (1 − Tr  𝑚)(1 − Tr  𝑚

)]
 𝑚  1 = Tr 𝑖,𝑗 𝑇𝑖𝑗|𝑟𝑖 𝑟𝑖|
Where 𝑇 is a 𝑚 × 𝑚 matrix such that:
• 𝑇 ≥ 0
• 𝑇𝑖𝑗 = 𝑟𝑖 𝑟𝑗 𝑟𝑖  𝑟𝑗
In order to compute the TFB we need to know: 𝑟𝑖, and 𝑟𝑖  𝑟𝑗
Computing the Truncated Fidelity Bounds (TFB)
cerezo@lanl.gov arXiv:1906.09253
T matrix is classically diagonalizable for small m!

6. Los Alamos National Laboratory
3/6/2020 | 6
Variational Quantum Fidelity Estimation (VQFE) Algorithm
1. First,  is diagonalized with a hybrid quantum-classical loop, outputting the largest
eigenvalues {𝑟𝑖} of , and a gate sequence that prepares the associated eigenvectors1.
2. Second, a hybrid quantum-classical computation gives the matrix elements of  in
the eigenbasis of : prepare superposition (|𝑟𝑖 + |𝑟𝑗 )/ 2, use SWAP Test.
3. Third, classical processing gives upper and lower bounds on 𝐹 ,  .
[1] R. LaRose, et al, “Variational quantum state diagonalization,” npj Quantum Information 5, 8 (2019).

7. Los Alamos National Laboratory
3/6/2020 | 7
Numerical Implementations
 Randomly chosen 
 Low rank 
 Computed TFB & SSFB
 Also provide Certified
Bounds (CB)
Sub- and super-fidelity bounds (SSFB)1: 𝐸 ,  ≤ 𝐹 ,  ≤ 𝐺 , 
𝐸 ,  = Tr  + 2[ Tr  2 − Tr ] and 𝐺 ,  = Tr  + (1 − Tr 2)(1 − Tr 2)]
Quantum Phase Transition:
 Thermal state of Ising chain of
N= 8 spins ½ in a transverse field.
 Computed TFB & SSFB.
 TFB: Better estimation of phase
transition location!
[1] J. Miszczak, et al, “Sub-and super-fidelity as bounds for quantum fidelity,” Quantum Inf & Comput 9, 103–130 (2009).

8. Los Alamos National Laboratory
3/6/2020 | 8
Complexity Analysis
• Problem: Low-rank Fidelity Estimation
Input: Two poly(n)-sized quantum circuit descriptions 𝑈 and 𝑉 that prepare
n-qubit states  and  on a subset of qubits all initially in the |0 state, and a
parameter 𝛾.
Promise: The state  is low rank (i.e., rank()= poly(n)).
Output: Approximation of 𝐹 ,  up to additive accuracy 𝛾.
The problem Low-rank Fidelity Estimation to within precision 𝛾 = 1/poly(n)
is DQC1-hard.
Low-rank Fidelity Estimation is classically hard!

9. Los Alamos National Laboratory
3/6/2020 | 9
Review of Main Results
Quantum Information Theory
• Novel upper and lower bounds for the Quantum Fidelity 𝐹 ,  base on the “Truncated Fidelity”.
Variational Hybrid Quantum-Classical Algorithm
• Variational Quantum Fidelity Estimation.
• Numerical implementations for low Rank states. We also heuristically show we need to consider a
number of eigenvalues in the truncated states which does not scale exponentially with n.
Complexity Analysis
• The problem of “Low Rank Fidelity Estimation” is DQC1-hard
cerezo@lanl.gov arXiv:1906.09253

10. Los Alamos National Laboratory
3/6/2020 | 10
Thank you for your attention!
See you all at the next APS March meeting!