Hybrid Quantum-Classical Hierarchy for Mitigation of Decoherence and Determination of Excited States

This paper shows that VQE is robust even in a parameterization-independent model in non-ideal conditions.

To understand this, the paper develops a theoretical model of variational state preparation called the variational channel state (VCS) preparation. VCS is just preparing an arbitrary pure state and then applying Kraus operators, so that we can study the optimal possible performance of HQC independently of ansatz choice or experimental apparatus. Okay I think this makes sense so far.

We want to find the minimum eigenstate $\ket{\psi}$ of Hamiltonian $H$ after VCS maps $\rho = \ket{\psi} \bra{\psi} \rightarrow \sum\_i K\_i \rho K\_i^\dagger$.

Writing VQE problem as trace of density matrix

In VQE we want to minimize $ H $. Since $H$ is a Hamiltonian which is Hermitian, we can decompose it as: $H = \sum\_E E \ket{E} \bra{E}$, so minimizing $H \ket{\psi}$ is the same as choosing $\ket{\psi} = \ket{E}$ for the $\ket{E}$ with the lowest eigenvalue.

Notice that \[ _{} ( H ) = _{} ( H ) = _{} H = E \]

Solving QCS VQE is the same as doing VQE on $H' = \sum\_i K\_i^\dagger H K\_i$

Doing VQE on QCS amounts to:

\[ _{} ( _i K_i K_i^H ) \]

We can use cyclic invariance of the trace to rewrite this as $\mathrm{Tr} \Big( \ket{\psi} \bra{\psi} H' \Big)$. Appendix D has a different approach which introduces Lagrange multiplier or something but I think this works too .

We know this can be rewritten as \[ _ H’ \]

Solving this quantifies optimal performance on VQE and determines state which achieves this regardless of ansatz.

Configuring quantum channels in this paper

This paper used 3 main quantum channels. Choosing the probabilities of certain errors was done in order to maintain two relations: \[ = = 0.05 \] where $T\_p$ is total experiment time, $T\_1$ is decay time, and $T\_2$ is dephasing time. These metrics are used widely in quantifying decoherence of a quantum processor. Enforcing this relation amounts to an experiment that uses $5\%$ of the time available (I think). I wonder what would happen if I vary this relation or even make them not equal?

Dephasing channel

The dephasing channel (denoted $F\_P (p) \[\rho \]$) has Kraus operators: \[ K_0 = I K_1 = Z \]

If we expand this out on a single-qubit density matrix $\rho$, we get \[ F_P(p) [] = ]

In this paper, we choose $p = -1 - \exp(-T\_p / T\_2)$ (for reasons I don’t understand yet) to get density matrix: \[ F_P(p) [] = ]

Amplitude and phase damping channel

The amplitude and phase damping channel (denoted $F\_{AP} (p) \[ \rho \]$) consists of an amplitude channel and dephasing channel applied independently to each qubit.

We can write $F\_{AP} (p) \[ \rho \] = F\_P (p\_1) \[ F\_A (p\_2) \[ \rho \] \]$ where $F\_P$ is defined above and $F\_A$ has Kraus operators: \[ K_0 = ]

Calculating the composite map on a single-qubit density matrix gives us: \[ F_{AP} (p) [ ] = ]

We choose $p\_2 = 1 - \exp (-T\_p / T\_1)$ and keep $p\_1$ the same as in Dephasing channel.

Depolarizing channel

The depolarizing channel (denoted $F\_D (p) \[ \rho \]$) has Kraus operators:

\[ K_0 = K_1 = X \]

\[ K_2 = Y \qquad K_3 = Z \]

Here we choose $p = 1 - \exp ( -T\_p / T\_2)$.

Understanding Figure 2

Here is Figure 2 from the paper. This is the main result about error-resistance of VQE.

Ntwali told me that the x-axis $R(A)$ is bond length in angstrom. The graph is supposed to be representing the fidelity of a 4-qubit quantum state representing $H\_2$ through several channels with and without variataional optimization.

I need to figure out how to decompose $H\_2$ into Hamiltonian form that depends on bond length… Maybe 1812.09976 can help?

Each channel has two lines associated with it. I’m thinking the solid line with markers is solving $\min\_{\ket{\psi}} \braket{ \psi \| H' \| \psi }$ while the markers only is solving $\min\_{\ket{\psi}} \braket{ \psi \| H \| \psi }$.

Then this solution is the initial state and we pass $\psi$ through the channel again to get the final state, and then compare fidelities.

Calculating Hamiltonian of $H_2$ based on intermolecular distance