Gate Set Tomography: Overcoming SPAM Errors

Hello! Welcome to our next lesson.

In our last session, we dove into Quantum Process Tomography (QPT) and saw how it could, in principle, fully characterize a quantum gate. However, we ended on a critical limitation: QPT's accuracy is fundamentally compromised by its assumption that state preparations and measurements (SPAM) are perfect. Any real-world SPAM error gets incorrectly attributed to the gate being characterized.

Today, we'll explore the solution to this problem: Gate Set Tomography (GST). Our goal is to analyze the principles of GST and explain how its clever use of "fiducial" and "germ" sequences allows it to simultaneously characterize the gates, the state preparation, and the measurement process, thereby overcoming SPAM errors. This self-consistent approach has become the gold standard for high-precision quantum hardware certification, as demonstrated in landmark experiments.

The Challenge of SPAM-Free Characterization

As we concluded last time, if you try to perform QPT on a gate and your initial state preparation is flawed (e.g., you create a state slightly off the intended Bloch sphere axis), QPT has no way of knowing. It sees the discrepancy in the final state and blames the gate. The same is true for measurement errors. This conflation of errors makes it impossible to say with high confidence whether you have a near-perfect gate with noisy SPAM, or a faulty gate with perfect SPAM.

GST solves this by redesigning the experiment from the ground up. Instead of characterizing a single gate in isolation, GST characterizes an entire set of operations—the gates, the initial state preparation (S for "state"), and the measurement (M for "measurement")—all at once. This is what we mean by "self-consistent."

The paper "Quantum logic with spin qubits..." which reported world-leading gate fidelities, explicitly relies on GST for this reason.

As the paper states, GST is a "self-consistent benchmarking technique" that, unlike QPT, is robust to SPAM errors. It does this by running long, structured sequences of gates and fitting one model to the entire dataset. This allows the model to distinguish errors intrinsic to a gate (which appear every time that gate is used) from errors specific to the start or end of a sequence (the SPAM).

The Anatomy of a GST Experiment

The power of GST comes from its unique experimental structure. Instead of simply applying a gate to a few input states, GST executes thousands of different, carefully constructed gate sequences. Each sequence has the following form:

Preparation Fiducial ($F_i$) → Germ Power ($g_k^L$) → Measurement Fiducial ($F_j$)

Let's break down what each of these components does.

This structure is also visualized nicely in the workflow diagram from the Nature paper:

• Fiducials ($F_i,F_j$): These are short gate sequences (e.g., X(π/2) or Y(π/2)) placed at the beginning and end of the main sequence. Their job is to prepare a wide range of input states and enable a wide range of measurements, effectively performing tomography on the central Germ Power block.
• Germs ($g_k$): These are also short gate sequences (e.g., a single gate like X(π/2) or a composite sequence like X(π/2)Y(π/2)). They are chosen specifically to be sensitive to different types of small errors in the gates.
• Germ Power ($g_k^L$): The germ is repeated $L$ times. This is the crucial step for amplifying tiny, coherent errors. A small error $ϵ$ in the germ $g_k$ becomes a much larger, more easily measurable error of approximately $Lϵ$ in the sequence $g_k^L$.

By running experiments for many combinations of fiducials $F_i,F_j$, germs $g_k$, and repetition lengths $L$, GST gathers comprehensive information about how all errors in the system compose and interact.

Deep Dive 1: Fiducials and Informational Completeness

The role of fiducials is to ensure that we can see the Germ Power block from all "angles." In QPT, we did this by preparing states like $|0⟩,|+⟩,|i⟩$, etc. GST does the same, but it doesn't assume the gates used to prepare these states are perfect.

The key requirement for the fiducials is that they generate state preparations and measurements that are informationally complete (IC).

Here's the core idea:

• An IC set of preparation states is a set of density matrices that can be linearly combined to form any density matrix. For a single qubit ($d=2$), this requires at least $d^2=4$ linearly independent states.
• Similarly, an IC set of measurements (represented by POVM effect matrices) must span the same space.
• GST uses fiducial sequences to achieve this. For example, starting from the machine's initial state ${\rho }_0$ (which is nominally $|0⟩⟨0|$), applying no gates (the empty sequence $\{\}$) prepares one state. Applying a $\pi /2$ X-gate, Gx, prepares another. The set of sequences {{}, Gx, Gy, GxGxGx, ...} produces a set of states.
• Crucially, GST does not assume that the Gx gate is a perfect X-rotation. It models the initial state as an unknown density matrix ${\rho }_0$ and the gates as unknown process matrices $G_x,G_y,\dots$. The "state" prepared by the fiducial Gx is $G_x({\rho }_0)$. The characterization of ${\rho }_0$ and $G_x$ are outputs of the GST analysis.

This is how GST defeats SPAM errors. An error in state preparation is just an imperfection in the model for ${\rho }_0$. An error in a measurement gate is an imperfection in the model for that gate, which is determined self-consistently with all its other uses in the experiment.

Deep Dive 2: Germs and Error Amplification

While fiducials provide the tomographic framework, germs are the precision tool for magnifying errors. Repeating a gate sequence makes tiny coherent errors accumulate until they are large enough to be detected above statistical noise.

However, as you know from your background in modeling physical systems, error behaviors can be complex. Simply repeating a single gate might not reveal all error types.

Let's summarize the physics of germ selection:

1. Simple Error Amplification: If a gate is supposed to rotate by $\theta$ but actually rotates by $\theta +ϵ$, repeating it $L$ times results in a rotation by $L\theta +Lϵ$. The error $ϵ$ is amplified by $L$.
2. Complex Errors (e.g., Tilt): If a gate is supposed to rotate by $\pi /2$ about the X-axis, but the axis is slightly tilted towards the Y-axis, this is a "tilt error." Simply repeating this gate might cause the error to cancel itself out. For example, the article shows that for a specific tilt error, four applications of the gate return you to the ideal operation, hiding the error.
3. Composite Germs: To detect these more subtle errors, we need germs composed of multiple different gates, like GxGy. This sequence is sensitive to a different combination of underlying error parameters than Gx or Gy alone.
4. Amplificational Completeness (AC): The goal is to choose a small set of germs that, collectively, amplify all possible physically distinct (gauge-invariant) errors in the gate set. Such a set is called amplificationally complete. The mathematical machinery of the Jacobian is used to find a set of germs whose sensitivities span the entire space of possible gate errors.

The Final Fit: Tying It All Together

After running thousands of sequences combining different fiducials, germs, and lengths, the final step is a large-scale optimization. A maximum-likelihood estimation algorithm finds the single model—comprising one initial state ${\rho }_0$, one measurement POVM $\{E_k\}$, and one process matrix for each gate in the set $\{G_m\}$—that best explains all the experimental data simultaneously.

Because a gate like Gx appears in many different contexts (as a fiducial, as a germ, inside other germs), its process matrix must consistently explain its behavior in all of them. This is what allows the fit to disentangle the gate's intrinsic error from the SPAM errors, which only occur at the very beginning or end of the sequences.

Conclusion

In this lesson, we have dissected the principles of Gate Set Tomography, the leading protocol for SPAM-free quantum device characterization.

Key takeaways:

• GST overcomes the primary limitation of QPT by being robust to State Preparation and Measurement (SPAM) errors.
• It achieves this by self-consistently characterizing the entire gate set, including the preparation and measurement operations, in a single comprehensive analysis.
• The experimental protocol relies on structured sequences of the form Fiducial - (Germ)^L - Fiducial.
• Fiducial sequences are chosen to be informationally complete (IC), providing a tomographically complete set of effective preparations and measurements.
• Germ sequences are chosen to be amplificationally complete (AC), ensuring that repetitions will amplify every possible type of coherent error in the gate set.
• By fitting a single model to a vast dataset of these structured experiments, GST can distinguish gate-intrinsic errors from SPAM errors, providing an unparalleled level of diagnostic detail and accuracy.

Preview of the next lesson:
GST provides the most detail possible about a gate set, but it is experimentally very demanding. Randomized Benchmarking (RB), which was mentioned as an alternative in the Nature paper, sits at the other end of the spectrum. It provides much less information—typically just a single number, the average gate fidelity—but is far more scalable and efficient to run. In our next lesson, we will derive the protocol for single-qubit randomized benchmarking and understand the theory that allows it to also be insensitive to SPAM errors.