Control, Readout, and Calibration

Control determines the fields acting on a quantum device; readout converts its response into a classical record; calibration estimates the parameters needed to make both reliable. The useful distinction is between the waveform requested, the evolution actually produced, and the information extracted. This page connects those physical processes, including reset and evolution during feedback delays.

Required background. Hardware Overview explains computational subspaces and physical operations; Metrics for Quantum Hardware distinguishes fidelity, assignment, and leakage; Quantum Instruments supplies conditional state updates.

Controlled dynamics and physical waveforms

A finite-dimensional model often begins with

$$H(t;\theta) = H_0(\theta) + \sum_{k=1}^{K} u_k(t)H_k(\theta),$$

where $\theta$ contains device parameters, $H_0$ is the drift Hamiltonian, $H_k$ are control generators, and $u_k(t)$ are physical fields or effective envelopes. For an open system with a suitable time-local description, the corresponding model may be

$$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H(t;\theta),\rho] + \mathcal D_{\theta,t}(\rho).$$

A Lindblad description with nonnegative instantaneous rates is one model for $\mathcal D_{\theta,t}$; it does not capture arbitrary environmental memory. Model the relevant extra modes explicitly when their history affects subsequent controls.

Real actuators constrain $u_k(t)$ through:

• amplitude and energy bounds;
• finite bandwidth and sample rate;
• phase, frequency, and timing resolution;
• slew-rate and smoothness limits;
• finite spatial resolution and addressing spillover;
• shared-resource and simultaneous-operation constraints;
• heating, scattering, photon-number, or motional limits;
• waveform memory and streaming limits.

A pulse that is optimal for an unconstrained model but cannot be generated, delivered, synchronized, or calibrated is not an implementable control.

Reachability and native operations

The available generators determine which transformations are reachable and how costly they are. In closed finite-dimensional systems, Lie-algebraic controllability tests can establish whether combinations of $iH_0$ and $iH_k$ generate the desired unitary algebra. This is an existence statement. It does not determine a robust, short, low-leakage waveform under experimental constraints.

A native gate specifies an interaction, computational subspace, pulse duration, rotating frame, and allowed simultaneous operations. Leakage and spectator effects depend on those choices. A virtual frame change alters the phases assigned to later pulses; it requires a consistent phase reference even when no physical pulse is applied. Pulse-Level Control explains the corresponding scheduling and waveform representation.

Delivered fields and causal response

The requested digital envelope $u_{\mathrm{cmd}}(t)$ generally differs from the field at the device. In a linear time-invariant approximation,

$$u_{\mathrm{dev}}(t) = \int_{-\infty}^{\infty} h(t-t') u_{\mathrm{cmd}}(t')\,dt'.$$

With the convention $U(\omega)=\int u(t)e^{-i\omega t}\,dt$, this becomes

$$U_{\mathrm{dev}}(\omega) = H(\omega)U_{\mathrm{cmd}}(\omega).$$

Causality requires $h(t)=0$ for $t<0$. A pure delay has $H(\omega)=e^{-i\omega\tau}$ and produces $u_{\mathrm{dev}}(t)=u_{\mathrm{cmd}}(t-\tau)$. A predetermined pulse can be launched early to compensate this delay; an unknown feedback decision cannot be sent before it exists.

$H(\omega)$ includes digital filters, converters, mixers or modulators, cables, amplifiers, resonances, reflections, optical elements, and device coupling. Predistortion attempts to choose $U_{\mathrm{cmd}}$ so that the delivered field matches a target. Exact inversion is unsafe near zeros of $H(\omega)$ and can amplify noise. A regularized frequency-domain inverse controls this amplification but does not by itself enforce causality, stability, finite pulse duration, or actuator bounds. In particular, a nonminimum-phase response need not have a stable causal inverse. van Dijk and colleagues relate control-electronics imperfections to qubit errors.

The linear model is only a starting point. Saturation, mixer imbalance, frequency-dependent phase, hysteresis, thermal effects, laser nonlinearities, ac Stark shifts, and state-dependent response can make the transfer context dependent.

Clock and reference integrity

Timing jitter, phase noise, frequency-reference drift, skew between channels, and trigger uncertainty can become quantum errors. A phase error $\delta\phi$ in a nominal equatorial qubit rotation changes its axis:

$$R_{\phi+\delta\phi}(\vartheta) \ne R_\phi(\vartheta).$$

Shared clocks reduce relative drift but can create common-mode correlated error. Independent references reduce some shared failure modes but require synchronization. The relevant specification follows from the quantum operation and circuit duration, not from a generic electronics data sheet.

Pulse synthesis and robustness

A control objective may minimize a cost

$$J[u] = 1-F_{\mathrm{target}}[u] + \lambda_E \sum_k \int_0^T \lvert u_k(t)\rvert^2\,dt + \lambda_L L[u] + \lambda_S S[u],$$

where $L[u]$ penalizes leakage and $S[u]$ penalizes bandwidth, roughness, sensitivity, or another engineering burden. The weights encode a design choice. A high simulated fidelity obtained by permitting excessive power or fragile spectral features may be inferior in the laboratory.

The penalty coefficients must give all terms compatible units. Gradient-based pulse search, analytic composite pulses, and experimental refinement address different information and cost constraints. Optimal Control develops the algorithms; the reviews by Glaser and colleagues and Koch and colleagues discuss their physical limitations.

Robust objectives

If uncertain parameters have distribution $\pi(\theta)$, an average robust objective is

$$\overline J[u] = \int J[u;\theta]\pi(\theta)\,d\theta.$$

A worst-case objective is

$$J_{\max}[u] = \max_{\theta\in\Theta} J[u;\theta].$$

Average robustness can tolerate rare poor regions; worst-case robustness can be conservative and expensive. State which uncertainty set or distribution was used and validate on held-out parameter values.

Leakage-aware shaping

Fast driving of a weakly anharmonic system can populate levels outside the computational subspace. Derivative-based quadrature corrections such as DRAG, spectral shaping, longer pulses, and optimal control can reduce leakage, but they require calibration of anharmonicity, transfer functions, and frame phases. Suppressing final leakage does not guarantee that transient leakage is harmless: population outside the code space can acquire phases or interact with spectators during the pulse.

Calibration as parameter estimation

A calibration experiment chooses settings $x$, observes data $D$, and estimates parameters $\theta$ through a model

$$p(D\mid x,\theta).$$

For example, a Rabi scan estimates an amplitude-to-rotation relation, a Ramsey scan estimates detuning and phase, and detector records estimate a classifier. Parameter uncertainty, model adequacy, and operating context determine whether a fitted value predicts another experiment. Quantum Measurement as Estimation explains likelihoods, bias, and uncertainty.

Calibration dependencies

Calibrations form a directed graph. A readout threshold may depend on resonance frequency and gain. A $\pi$-pulse amplitude may depend on drive frequency, transfer function, and pulse duration. An entangling gate may depend on both one-qubit frames and coupler bias. If an upstream parameter changes, downstream calibrations may become stale.

For a dependency graph $G_{\mathrm{cal}}=(V,E)$, an edge

$$a\longrightarrow b$$

means calibration $b$ assumes a valid result from $a$. A directed acyclic portion can be updated in topological order. Real systems may contain loops, such as readout needed to calibrate control while control is needed to prepare readout references. Those loops require bootstrapping, joint estimation, or progressively refined procedures.

The physical reason for these dependencies matters: changing amplifier gain changes the detector distributions, while changing drive frequency can alter a pulse’s rotation axis and rate. Record units and correlations between fitted parameters, then recheck the affected predictions. Kelly and colleagues describe the graph approach; Calibration Loops develops drift tracking and repeated experimental updates.

Open-loop, closed-loop, and adaptive calibration

Model-based open-loop control computes a waveform from an estimated model and executes it without measuring during the operation. It can still be recalibrated between experiments.

Closed-loop calibration evaluates an outcome and updates control parameters over repeated trials. The feedback acts on future experiments, not necessarily within one quantum trajectory.

Real-time measurement feedback uses a record from the current shot to choose a later operation in that same shot.

These loops have different latency and disturbance requirements. Calling all of them “feedback” without a timescale is ambiguous.

Rabi and Ramsey calibration

Rabi amplitude calibration

For a resonantly driven ideal two-level system initially in $\lvert0\rangle$,

$$P_1(t) = \sin^2 \left( \frac{\Omega t}{2} \right).$$

A nominal $\pi$ pulse satisfies

$$t_\pi = \frac{\pi}{\Omega}.$$

In practice, fit contrast, offset, damping, detuning, and possibly leakage. A Rabi scan estimates an amplitude-times-duration relation at one operating point. It does not by itself calibrate phase, simultaneous context, or long-sequence coherent accumulation.

Ramsey frequency and frame calibration

Fix $Z=|0\rangle\langle0|-|1\rangle\langle1|$ and the physical qubit Hamiltonian $H_q=-\hbar\omega_q Z/2$. In a frame rotating at the drive frequency, use the detuning convention

$$\delta=\omega_{\mathrm{drive}}-\omega_q, \qquad H_{\mathrm{rot}}=\frac{\hbar}{2} \left[\delta Z+\Omega(\cos\phi X+\sin\phi Y)\right].$$

Here $\Omega$ and $\delta$ are angular frequencies and $\phi$ defines the drive axis in this frame. During free evolution, $U(\tau)=e^{-i\delta\tau Z/2}$. Starting in $|0\rangle$, apply an ideal $R_x(\pi/2)$ pulse, wait $\tau$, then apply $R_\phi(\pi/2)$. Direct multiplication gives

$$P_1(\tau,\phi)=\frac{1+\cos(\delta\tau-\phi)}{2}.$$

The $\phi=0$ fringe is even in detuning: its frequency alone does not identify the sign. Measuring also $\phi=\pi/2$ yields the sine quadrature and distinguishes the sign locally. Samples at equally spaced delays still alias frequencies separated by $2\pi/\Delta\tau$; a bounded prior range or additional delay choices are needed. Contrast loss, offsets, and phase uncertainty must be included in a realistic fit.

With this convention, a measured positive $\delta$ calls for reducing the carrier frequency by $\delta$. Choosing the analysis phase $\phi=\delta\tau$ can instead compensate a known idle phase. This frame tracking does not remove the population error of an off-resonant finite pulse. For a square pulse of fixed axis,

$$P_1(t)=\frac{\Omega^2}{\Omega^2+\delta^2} \sin^2\!\left(\frac{\sqrt{\Omega^2+\delta^2}\,t}{2}\right),$$

whose maximum is below one when $\delta\ne0$. Rabi and Ramsey Control derives the driven dynamics.

Testing predictions beyond the fit

Rabi data used to choose a pulse do not independently establish its gate fidelity. Test new sequences, delays, input states, and simultaneous operations after fixing the parameters. Long sequences expose coherent accumulation that one population scan may miss; randomized benchmarking and tomography answer different questions, as discussed in Metrics for Quantum Hardware.

Initialization and reset

A unitary acting on a closed device preserves the eigenvalues and von Neumann entropy of its state. It cannot map every unknown input to the same pure state. Reset requires exporting information or entropy to something else: a reservoir, discarded auxiliary system, or measurement record. Coherent transfer to a fresh pure auxiliary mode can reset the device while moving its former state to that mode; preparing the auxiliary is part of the physical cost.

Passive relaxation, optical pumping, engineered dissipation, and measurement followed by conditional correction implement different reset channels. Heralded preparation instead specifies a conditional output and a success probability. Compare duration, unconditional residual excitation, leakage or loss, and disturbance of neighboring systems. Excellent conditional fidelity alone does not give a useful repetition rate.

Ristè and colleagues demonstrate measurement-based reset. The delay example below explains why even an accurately classified initial state may require a different correction by the time the pulse arrives.

Readout records and conditional states

A quantum observable couples to a pointer; the pointer produces a field, fluorescence, charge response, or detector event; collection and electronics produce a record. Filtering or classification then assigns an outcome. The quantum instrument describes this entire outcome-conditioned physical process, including its backaction, rather than an extra operation performed after classification.

Filtered records and classifiers

Represent the sampled record as a real vector $y$; for complex IQ data, stack its real and imaginary components. Suppose the two classes are Gaussian,

$$y\mid j\sim\mathcal N(\mu_j,C),\qquad j=0,1,$$

with the same positive-definite covariance $C$ and known prior probabilities $\pi_j>0$. Minimizing the probability of a wrong label gives the log-likelihood-ratio test

$$\begin{aligned} w&=C^{-1}(\mu_1-\mu_0),\\ \text{report }1\quad&\text{if}\quad w^T y> \frac12(\mu_1+\mu_0)^Tw+\log\frac{\pi_0}{\pi_1}. \end{aligned}$$

Subtracting the two Gaussian quadratic forms derives this rule: the common $y^TC^{-1}y$ term cancels. White noise gives a filter proportional to the difference of mean records. Unequal covariances generally give a quadratic decision rule; unequal error costs also change the threshold. A complex scalar cannot be ordered by an ordinary real threshold without choosing a real projection or another explicit rule.

Relaxation during measurement changes the distribution of temporal records. Gambetta and colleagues compare linear and nonlinear filters for this problem. Test assignment on independent records; adjusting the threshold on the same records used to estimate its error produces an optimistic result.

POVMs and instruments

The detector outcomes are represented by a POVM $\{M_m\}$:

$$p(m\mid\rho) = \operatorname{Tr}(M_m\rho).$$

The state update is represented by a quantum instrument $\{\mathcal I_m\}$:

$$\widetilde\rho_m = \mathcal I_m(\rho), \qquad p(m\mid\rho) = \operatorname{Tr}\widetilde\rho_m.$$

The maps $\mathcal I_m$ are completely positive and their sum is trace preserving when every outcome is retained. Their effects satisfy $M_m=\mathcal I_m^\dagger(I)$. Two instruments can have identical effects and different conditional states. The normalized state is $\widetilde\rho_m/p(m\mid\rho)$ for a nonzero outcome probability.

Quantum Instruments develops this distinction; Measurement Tomography explains effect reconstruction. SPAM Errors discusses the ambiguity between imperfect reference preparation and imperfect readout. Correcting terminal assignment statistics, as in Measurement Error Mitigation, does not undo the measurement backaction.

Quantum non-demolition behavior

A measurement is quantum non-demolition relative to an observable when repeated measurement can reveal that observable without the measurement dynamics causing transitions between its eigenspaces, under the stated model. Practical tests should separate:

• first-readout assignment error;
• conditional state disturbance;
• relaxation or excitation between readouts;
• leakage and loss;
• correlated detector memory.

High agreement between repeated labels can arise from a detector bias and is therefore not sufficient alone.

Multiplexing and readout crosstalk

Multiplexing shares detectors, frequency bands, optical paths, cameras, amplifiers, or digitizers. It reduces hardware fan-out but introduces dynamic-range, spectral-collision, bandwidth, and classifier-correlation constraints. Compare isolated and simultaneous readout, including spectator dephasing and measurement-induced transitions.

The full confusion matrix can grow exponentially with qubit count, so scalable models often assume locality or low-order correlations. Those assumptions should be tested rather than treated as definitions.

Feedback delay and conditional correction

Choose an explicit boundary for each map. Let $\mathcal I_m$ describe the measurement through the time its output signal becomes available. Let $\mathcal D_m$ describe the subsequent processing and waiting interval, and let $\mathcal C_m$ describe the correction, including evolution during its finite pulse. Then

$$\mathcal E_{\mathrm{fb}}(\rho) =\sum_m\mathcal C_m\circ\mathcal D_m\circ\mathcal I_m(\rho).$$

If $\sum_m\mathcal I_m$ is trace preserving and each subsequent map is a channel, this is a channel. An initially correlated or memory-bearing environment may require an enlarged state to justify those composed maps. Discarding some outcomes instead gives a trace-decreasing accepted branch.

Measurement, propagation, digitization, inference, and actuation all take time, but count each interval once. For serial stages their durations add. Overlapping stages require the actual dependency schedule. Decoherence already included in $\mathcal I_m$ must not be inserted again as an extra idle channel. Feedback from Records treats conditional estimation and delayed control; Mid-Circuit Measurement and Feedforward explains the corresponding circuit branches.

For a simple reset example, suppose a perfect instantaneous $Z$ measurement reports the initial state, followed by a delay $\tau$ with amplitude damping and an instantaneous $X$ correction only when the result was 1. If the initial excitation probability is $p_1$, then

$$p_{\mathrm{exc,final}}=p_1\left(1-e^{-\tau/T_1}\right).$$

An initially excited qubit that survives the delay is reset correctly. One that decays before the correction is excited again by that correction. Accurate initial assignment therefore does not eliminate latency error. This model deliberately excludes imperfect readout, excitation from the bath, and finite correction duration.

The correction can instead use the predicted state at actuation time. Conditional on the recorded result 1, the excitation probability is $q=e^{-\tau/T_1}$. Doing nothing leaves excitation $q$; applying $X$ leaves $1-q$. Therefore apply $X$ only when $q>1/2$, or $\tau<T_1\ln2$, and otherwise do nothing. The recorded-0 branch remains in the ground state. At a fixed delay, the smallest final excitation probability among these two actions is

$$p_{\mathrm{exc,opt}} =p_1\min\!\left(e^{-\tau/T_1},1-e^{-\tau/T_1}\right).$$

This optimizes one final ideal correction, without another measurement or reset channel; it is not a recommendation to wait longer. With imperfect readout or heating, predict $q_m=P(1\text{ at actuation}\mid m)$ from the conditional dynamics before choosing the action. The comparison of $q_m$ and $1-q_m$ still applies to ideal $I/X$ corrections. Feedback from Records develops the state-estimation viewpoint.

Crosstalk and simultaneous control

For small classical cross-coupling, delivered controls may be approximated by

$$\mathbf u_{\mathrm{dev}}(\omega) = H(\omega)\mathbf u_{\mathrm{cmd}}(\omega),$$

where off-diagonal entries of $H$ represent channel crosstalk. Predistortion can compensate a stable, identified matrix over a limited bandwidth. It cannot remove state-dependent interactions, heating, shared quantum modes, detector backaction, or unmodeled nonlinearities.

Simultaneous calibration should probe:

• neighboring and distant spectator states;
• overlapping gates and measurements;
• power and duty-cycle dependence;
• frequency and timing collisions;
• correlated residuals rather than only marginal errors;
• context changes caused by routing or compiler schedules.

An isolated calibration table is not sufficient evidence for a parallel processor.

Drift and repeated calibration

A fitted parameter $\widehat\theta(t_j)$ describes a device near its acquisition time and operating conditions. Temperature changes, loading, gain drift, and neighboring pulses can change the prediction. Repeating short diagnostic experiments tests whether the same model still fits; changing a parameter in response to every noisy fluctuation can itself destabilize the controls. Calibration Loops discusses tracking, false alarms, and the tradeoff between estimation noise and drift.

Scaling also constrains how fast records can be processed. If each error-correction round produces $B$ bits every $t_{\mathrm{round}}$, the raw syndrome rate is $B/t_{\mathrm{round}}$. Sustained throughput must keep up with this stream, while latency must meet the deadlines of operations that depend on a decoded result. Pipelining can satisfy throughput without reducing the latency of one result. Decoders develops these distinctions.

Worked pulse, readout, and timing example

Consider an ideal qubit driven on resonance with observed Rabi frequency

$$f_R = 10\,\mathrm{MHz}.$$

Since $\Omega=2\pi f_R$, a $\pi$ pulse has duration

$$t_\pi = \frac{\pi}{2\pi f_R} = \frac{1}{2f_R} = 50\,\mathrm{ns}.$$

A two-quadrature Ramsey fit with the convention above finds $\delta/(2\pi)=200\,\mathrm{kHz}$. Lowering the drive frequency by $200\,\mathrm{kHz}$ restores resonance. Tracking the idle phase $\delta t$ is useful for scheduling, but does not substitute for this frequency correction during a finite pulse. Test the corrected control with new delays and longer sequences.

Suppose the readout statistic is Gaussian with equal variance:

$$\begin{aligned} z\mid0 &\sim \mathcal N(-1,0.5^2), \\ z\mid1 &\sim \mathcal N(1,0.5^2). \end{aligned}$$

With equal prior probabilities and threshold $z=0$, each idealized assignment error is

$$\Phi(-2) \approx 0.0228,$$

so the mean assignment fidelity is about $97.7\%$. This calculation assumes stationary Gaussian distributions and no relaxation, leakage, or drift. Those assumptions are tested on held-out records.

Finally, let the measured feedback stages be

$$\begin{aligned} t_{\mathrm{int+prop}} &=200\,\mathrm{ns}, \\ t_{\mathrm{dig+infer}} &=70\,\mathrm{ns}, \\ t_{\mathrm{decide+trigger}} &=80\,\mathrm{ns}, \\ t_{\mathrm{actuate}} &=50\,\mathrm{ns}. \end{aligned}$$

The total is

$$t_{\mathrm{fb}} = 400\,\mathrm{ns}.$$

Suppose the measurement instrument includes the first $200\,\mathrm{ns}$, subsequent waiting includes $70+80=150\,\mathrm{ns}$, and the correction map includes the final $50\,\mathrm{ns}$. These maps already cover the full $400\,\mathrm{ns}$; appending another $400\,\mathrm{ns}$ damping channel would count the time twice. Quoting only classifier time would instead omit real evolution.

Exercises

1. Calibrate a π pulse

An ideal resonant Rabi experiment gives $f_R=8\,\mathrm{MHz}$. Find the $\pi$-pulse and $\pi/2$-pulse durations.

Solution

For $P_1(t)=\sin^2(\pi f_R t)$,

$$t_\pi = \frac{1}{2f_R} = \frac{1}{16\,\mathrm{MHz}} = 62.5\,\mathrm{ns}.$$

The $\pi/2$ pulse has half that duration:

$$t_{\pi/2} = 31.25\,\mathrm{ns}.$$

This conversion assumes amplitude scales linearly, the drive is resonant, and pulse edges or transfer distortion do not alter the effective rotation.

2. Determine the Ramsey sign

With the pulse convention above and $\delta\tau=\pi/3$, find $P_1$ for analysis phases 0 and $\pi/2$. How do these change if detuning changes sign?

Solution

For phase 0, both signs give $P_1=(1+\cos(\pi/3))/2=3/4$. For phase $\pi/2$, positive detuning gives $(1+\sqrt3/2)/2\approx0.9330$ and negative detuning gives $(1-\sqrt3/2)/2\approx0.0670$. The sine quadrature resolves this sign ambiguity, though equally spaced delay samples still have frequency aliases.

3. Propagate a pure delay

A delivery line has transfer function

$$H(\omega) = e^{-i\omega\tau}$$

with $\tau=12\,\mathrm{ns}$. What time-domain effect does it produce, and how should a synchronized schedule compensate it?

Solution

The inverse transform gives a delayed waveform:

$$u_{\mathrm{dev}}(t) = u_{\mathrm{cmd}}(t-\tau).$$

For a waveform known in advance, the controller can launch that channel $12\,\mathrm{ns}$ earlier relative to a common device-time reference. An unknown measurement-dependent command must still incur the physical propagation delay. Compensation must include every channel’s delay; advancing one waveform without updating triggers and phase references can create a different timing error.

4. Regularize an inverse filter

Why is the formal predistortion

$$U_{\mathrm{cmd}}(\omega) = \frac{U_{\mathrm{target}}(\omega)} {H(\omega)}$$

unsafe when $\lvert H(\omega)\rvert$ is very small?

Solution

Division by a small transfer magnitude produces a very large command amplitude and amplifies measurement noise and modeling error. The requested signal may violate actuator limits or excite unmodeled dynamics. A practical inverse limits bandwidth, penalizes command power, or uses a regularized factor such as

$$\frac{H^\ast(\omega)} {\lvert H(\omega)\rvert^2+\lambda}.$$

The achieved device waveform and gate must then be validated experimentally.

5. Include unequal readout priors

Let $z\mid0\sim\mathcal N(-0.8,0.4^2)$ and $z\mid1\sim\mathcal N(0.8,0.4^2)$. Find the threshold minimizing the total assignment error for equal priors, then for $\pi_0=0.9$, $\pi_1=0.1$.

Solution

The scalar threshold is

$$z_c=\frac{\mu_0+\mu_1}{2} +\frac{\sigma^2}{\mu_1-\mu_0}\log\frac{\pi_0}{\pi_1}.$$

Equal priors give $z_c=0$ and error $\Phi(-2)\approx0.0228$. Unequal priors give $z_c=0.1\log9\approx0.2197$. The total error becomes

$$0.9\left[1-\Phi\!\left(\frac{z_c+0.8}{0.4}\right)\right] +0.1\Phi\!\left(\frac{z_c-0.8}{0.4}\right) \approx0.0122.$$

The less frequent class now has a larger conditional error; the weighted total is smaller. A classifier tuned to these priors is not automatically optimal for a different input ensemble.

6. Reset by coherent transfer

A device has an arbitrary state $\rho$, and an auxiliary qubit starts in $|0\rangle$. What does a SWAP do to the two reduced states? Does it remove the joint entropy?

Solution

$$\operatorname{SWAP} \left(\rho\otimes|0\rangle\langle0|\right) \operatorname{SWAP}^\dagger =|0\rangle\langle0|\otimes\rho.$$

The device is reset and the auxiliary inherits its state. The joint entropy remains $S(\rho)$. Reusing the auxiliary requires another reset process; a unitary on the device alone could not perform this operation for every $\rho$.

7. Reset with a stale measurement result

Under the ideal delayed-reset model above, take $p_1=1/2$, $T_1=20\,\mu\mathrm{s}$, and $\tau=400\,\mathrm{ns}$. Find the final excitation probability and compare it with the zero-delay result. Is the conditional $X$ still the better action? Repeat that comparison at $\tau=T_1$.

Solution

Only the initial excited branch can fail. Its decay probability during the delay is $1-e^{-0.02}$, after which the conditional $X$ pulse re-excites the qubit. Thus

$$p_{\mathrm{exc,final}}=\tfrac12(1-e^{-0.02}) \approx0.00990.$$

The result tends to zero as $\tau\to0$ and to $1/2$ as $\tau/T_1\to\infty$. For long delay, the initial outcome is a poor guide to the state just before correction.

Here $q=e^{-0.02}>1/2$, so the conditional $X$ is optimal among the two allowed actions. The decision changes at $T_1\ln2\approx13.9\,\mu\mathrm{s}$. At $\tau=T_1$, the fixed conditional-$X$ rule leaves excitation $(1-e^{-1})/2\approx0.316$, whereas doing nothing leaves $e^{-1}/2\approx0.184$. Relaxation has made the ground state more likely even on the branch initially measured as excited.

8. Compare average and worst-case robust control

Two uncertain parameter values have costs $J_1=10^{-4}$ and $J_2=10^{-2}$ with probabilities $0.99$ and $0.01$. Compute the average cost and worst-case cost.

Solution

$$\begin{aligned} \overline J &= 0.99(10^{-4}) + 0.01(10^{-2}) \\ &= 1.99\times10^{-4}. \end{aligned}$$

The worst-case cost is

$$J_{\max} = 10^{-2}.$$

The average objective regards the poor region as rare; the worst-case objective does not. The appropriate choice depends on whether rare excursions can be detected, tolerated, or made safe.

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