Christopher E. Granade

Centre for Engineered Quantum Systems
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This talk can be summarized in one slide.

\begin{equation} \Pr(\text{model} | \text{data}) = \frac{\Pr(\text{data} | \text{model})}{\Pr(\text{data})} \Pr(\text{model}) \end{equation}

Bayesian methods for experimental data processing *work*.

How do we assess if quantum computing is possible, given an experimental demonstration of a prototype?

(Classical computing is impossible.)

Colloquially, a classical computer is something that acts in most ways and for most of the time like we would expect a classical computer to act.

Classical computing models *preclude* some error
sources (comets, the NSA, etc.).

Take-away for quantum applications: we need to make and state assumptions, and to qualify our conclusions.

To a Bayesian, this means we need to be careful to record our priors.

(Frequentism uses priors, too: they're called *models*.)

What models and priors should we use to evaluate QIPs?

Born's rule gives predictions in terms of a state $\rho$, \[ \Pr(E | \rho) = \Tr[E \rho] \] for a measurement effect $E$.

This is a likelihood, so use Bayes' rule to find the state $\rho$. [Jones 1991, Blume-Kohout 2010]

Numerical evaluation is a kind of *classical simulation*.

We use sequential Monte Carlo (SMC) to compute Bayes updates.

\begin{align} \Pr(\rho) & \approx \sum_{p\in\text{particles}} w_p \delta(\rho - \rho_p) \\ w_p & \mapsto w_p \times \Pr(E | \rho_p) \end{align}

Uses simulation as resource.

Gives estimates of posterior over states $\rho$. Lots of advantages:

- Region estimation [Ferrie 2014]
- Model selection and averaging

[Wiebe*et al*2014, Ferrie 2014] - Hyperparameterization [Granade
*et al*2012]

```
import qinfer as qi
basis = qi.tomography.gell_mann_basis(3)
prior = qi.tomography.GinibreDistribution(basis)
model = qi.BinomialModel(qi.tomography.TomographyModel(basis))
updater = qi.smc.SMCUpdater(model, 2000, prior)
heuristic = qi.tomography.rand_basis_heuristic(updater, other_fields={
'n_meas': 40
})
```

How do we encode the assumption that the state $\rho$ is near a state $\rho_\mu$?

We use amplitude damping to contract uniform prior $\pi(\rho)$ [Granade and Combes (upcoming)].

Choose $\rho_*$, $\alpha$, $\beta$ such that $\expect[\rho_{\text{sample}}] = \rho_\mu$ and $\expect[\epsilon]$ is minimized.

Posterior covariance characterizes uncertainty.

Example: depolarizing channel dominates uncertainty in our QPT simulation.

(Useful in space.)

We don't need to assume that the "true" state follows a random walk.

Can also consider other models.

\[ U(t) = \ee^{-\ii H t} \]

Does the device evolve under the correct $H$? Does a simulator implement the correct $H$?

As before, we can answer this using simulation as a resource.

- Classical resources (particle filters, rejection sampling).

[Granade*et al*2012] - Quantum resources (semiquantum particle filters).

[Wiebe*et al*2014] - Small quantum resources (quantum bootstrapping).

[Wiebe*et al*2015]

The prior $\pi(H)$ again encodes the assumptions that we make about $H$.

\begin{align} H(\vec{x}) & = \sum_{\langle i, j \rangle} x_{i,j}\,\sigma_z^{(i)} \sigma_z^{(j)} \\ H(\vec{x}) & = \sum_{i,j,\alpha,\beta} x_{i,j,\alpha,\beta}\,\sigma_{\alpha}^{(i)} \sigma_{\beta}^{(j)} \end{align} Can also include Hermitian generators as well as skew-Hermitian. \begin{align} L(\omega, T_2) & = -\ii \omega (\id \otimes \sigma_z - \sigma_z^\TT \otimes \id) + \frac{1}{T_2} \left( \sigma_z^\TT \otimes \sigma_z - \id \otimes \id \right) \end{align}

Model selection provides a way to test assumptions by progressively considering more general models.

AIC, ABIC, Bayes factor, model-averaged estimation, etc.

\begin{align} \text{BF}(A : B) & \defeq \frac{\Pr(A | \text{data})}{\Pr(B | \text{data})} = \frac{\Pr(\text{data} | A) \Pr(A)}{\Pr(\text{data} | B) \Pr(B)} \end{align}

\[ \Pr(\text{O-beam} | A, b, \phi; \theta) = \frac{1 + b}{2} + \frac{A}{2} \cos(\phi + \theta) \]

Compare models $\phi \ne \phi(t)$ and $\phi$ following a Markovian Gaussian process.

We analyze the same data with both models, recording the total likelihood as we go.

Using Bayes factor analysis, we can conclude that the drifting model better explains the observed data.

Using Bayesian inference, Hamiltonian learning
can proceed *adaptively*.

Necessary [Hall and Wiseman 2012] and sufficient [Ferrie *et al* 2013] for exponential improvement
in the number of experiments required to learn Hamiltonians.

For building algorithms, more convenient to use circuits than continuous-time evolution.

Can enforce the gate model to some degree with how we design our pulses.
We can use more general simulation methods (e.g.: cumulant [Cappellaro *et al* 2006]) to reason about
validity of the gate model.

Same techniques as before apply to enable learning *snapshots* of dynamics.
Choi-Jamiołkowski isomorphism lets us rewrite process tomography as (ancilla-assisted)
state tomography.

\[ \Tr[E \Lambda(\rho)] = \Tr[(\id \otimes E) J(\Lambda) (\rho^\TT \otimes \id)] = \sbraket{\rho^\TT, E | J(\Lambda)} \]

This is difficult: limited to sampling statistics, no exponential improvement from longer evolutions.

Want something more like Hamiltonian learning. Recent example: gate set tomography [Blume-Kohout *et al* 2013].
GST is in turn difficult due to parameter count.

What do we actually *want* out of our gate model?

Common to describe gate models in terms of the *average gate fidelity*.
If errors are not strongly gate-dependent, also common to quote AGF averaged
over gates.

For some models, AGF can describe error correcting thresholds. More generally,
unreasonably strong assumptions may be needed [Sanders *et al* 2015].

All the same, let's keep searching under the streetlight, keeping track of our assumptions.

By choosing random sequences $\{U_1, U_2, \dots\}$,
can implement twirling superchannel $W[\cdot]$
with *imperfect* gates.

Marginalizing over sequence choice removes quantum simulation, yielding simple (scalar!) algebraic models.

\begin{align} \Pr(\text{survival} | A, B, p; m) & = A p^m + B \\ A & \defeq \Tr(E \Lambda[\rho - \id / d]) \\ B & \defeq \Tr(E \Lambda[\id / d]) \\ p & \defeq \frac{dF - 1}{d - 1} \end{align}

We now have a *simulation-free* likelihood function
that assesses average gate fidelity.

This is ideal for *embedded* applications.

Rejection sampling with Gaussian resampling allows for constant-memory embedded Bayesian inference [Wiebe and Granade (upcoming)].

Less general than SMC, but significantly faster and more parallelizable. Ideal for FPGA-based estimation of randomized benchmarking model.

Modern experiment control via FPGAs, so embed inference directly in control hardware to make an online fidelity oracle.

We have lots of prior information: each pulse is a pertubation of a pulse we've already characterized.

Since we're being honest frequentists (*aka* Bayesians),
let's use that to accelerate estimation.

With lots of data, least-squares fitting works fine, but it isn't stable with small amounts of data. Bayesian inference is nearly optimal for the strong prior.

Only remaining simulation cost: reduction of *pulses*
to *gates*.

...but nature does this for us.

Use array of quantum coprocessors to evaluate pulses in parallel, for different hypotheses about the true evolution. [Granade 2015]

How do we *make* a useful quantum device?

Treat characterization parameters as *fitness functions*.

Optimal control using AGF $F$ as a fitness has been applied very successfully to design pulses.

Khaneja derivative allows for using gradient-ascent methods to optimize $F$.

- Leakage
- Unitarity
- Haas-Puzzuoli decoupling fitness

Several of these fitness functions can also be experimentally measured using randomized benchmarking.

- $F$: interleaved 2-design twirl
- Leakage: 1-design twirl
- Unitarity: 2-design twirl on two copies

Using the same techniques, then, we can embed fidelity, leakage and unitarity estimation all into control hardware.

Immediately allows embedding RB-based pulse optimization.

- Ad-HOC / ORBIT [Egger and Wilhelm
*et al*2014, Kelly*et al*2014] - ACRONYM [Ferrie and Moussa 2014]

Multi-objective genetic algorithms can be used to design pulses that work over a range of models.

- Memetic step
- ACRONYM / SPSA
- Evolutionary Strategies
- Mutation rate, memetic paramteters co-evolve
- Tournaments
- Randomly selected from ensemble of tournament fns.

MOQCA is *black-box* for its oracle. Robust against
noisy oracles.

Not the most efficient, but could possibly work in much larger systems and with other kinds of distortions.

We consider controls up to $70$ MHz, but we want robustness to $\pm 100$ kHz static field, ringdown distortion, imperfect measurement of distortion.

Memetic optimization finds Pareto optimal pulses with fidelity $\ge 0.99$.

200 generations, 140 individuals, 5 hypotheses ($\pm 100 \text{kHz}$), noisy evaluation of distortion (~20 dB SNR)