Separate and efficient characterization of state-preparation and measurement errors using single-qubit operations

This paper presents an efficient protocol using only single-qubit gates and repeated non-destructive measurements to separately characterize state-preparation and measurement errors, demonstrating its application on IBM Quantum devices and highlighting the necessity of accounting for state-preparation errors to avoid biased results in measurement-error mitigation.

The Gist

Imagine you are trying to take a very precise photograph of a tiny, fragile object (a quantum bit, or "qubit"). To get a good picture, you need two things to work perfectly:

1. Setting the scene (State Preparation): You must place the object in the exact right position before you snap the photo.
2. Taking the photo (Measurement): Your camera must record exactly what is there without blurring or misinterpreting the image.

In the world of quantum computers, both of these steps are prone to errors. Often, the mistakes happen before the computer even starts its real work (setting the scene wrong) or after it finishes (the camera misreading the result). These are collectively called SPAM errors (State-Preparation And Measurement errors).

The problem is that most existing methods for fixing these errors treat them as a single, messy blob. They assume the "camera" is the only thing going wrong, or they try to fix everything at once using complex, slow, and error-prone tools.

This paper introduces a new, clever method called QSPAM (Quantum SPAM) that acts like a detective, separating the "setting the scene" errors from the "taking the photo" errors using only simple, fast tools.

The Core Idea: The "No-Reset" Trick

Usually, when you measure a quantum bit, the process destroys the state, and you have to start over from scratch to try again. This paper proposes a different approach: measure the same qubit twice in a row without resetting it.

Think of it like this:

• Standard Method: You ask a friend, "Is the light on?" They say "Yes." You then reset the room, ask again, and they say "No." You have to guess if the light changed, or if your friend is just bad at answering.
• QSPAM Method: You ask, "Is the light on?" They say "Yes." Without changing the room, you immediately ask, "Is the light still on?" They say "Yes."

By looking at the pattern of answers from these back-to-back questions, the authors show you can mathematically untangle the two problems:

1. Did the friend start with the light actually off, but think it was on? (State Preparation Error)
2. Did the friend see the light correctly but accidentally say the wrong word? (Measurement Error)

How They Did It (The Simple Tools)

The authors didn't need complex, heavy machinery. They used only single-qubit operations (simple rotations of the quantum bit) and repeated measurements.

• The Analogy: Imagine trying to calibrate a scale that is both unbalanced (it starts with a weight on it) and has a sticky needle (it doesn't always point to the right number). Instead of building a new, expensive scale, you just put a known weight on it, weigh it, then weigh it again immediately. By comparing the two results, you can calculate exactly how much the scale was off at the start versus how much the needle is sticking.

What They Found

The team tested this on real quantum computers provided by IBM. Here is what they discovered:

1. The Errors are Real and Separate: They found that "setting the scene" errors (preparation) and "reading the result" errors (measurement) are distinct. In some cases, the preparation was off by up to 6.5%, and the reading errors were as high as 19%. That is a huge amount of noise for a computer trying to do precise math.
2. The "Camera" Isn't Always Simple: They found that for some qubits, the measurement process is more complex than a simple "yes/no" switch; it has a bit of a "glitch" that makes it behave in a non-standard way. Their new protocol could detect this, whereas older methods would have missed it.
3. Fixing Only Half the Problem Makes It Worse: This is a crucial finding. If you try to fix the "camera" errors (measurement) but ignore the "setting the scene" errors (preparation), your final answer isn't just slightly wrong—it can be wildly wrong.
   • The Metaphor: Imagine you are trying to calculate the average height of a group of people. If you use a ruler that is bent (measurement error), you get a wrong answer. But if you also put everyone on a platform that is tilted (preparation error) and only try to fix the ruler, your final calculation might end up saying people are 10 feet tall! The paper shows that ignoring the "tilted platform" leads to "non-physical" results (numbers that don't make sense in reality).

Why This Matters

The paper argues that for quantum computers to be useful, we need to know exactly where the errors are coming from.

• Efficiency: Their method is fast. It doesn't require building complex circuits that grow with the size of the computer. It works just as well for 2 qubits as it does for 100.
• Accuracy: By separating the errors, they can fix them individually. This leads to much more accurate results when running quantum algorithms.
• Reality Check: They proved that the "standard" way of fixing errors (which assumes the setup is perfect) is often lying to us, giving us confidence in wrong answers.

In short, the authors built a simple, efficient "diagnostic tool" that tells quantum engineers exactly how their machine is messing up the setup and the reading, allowing them to fix the machine properly rather than just guessing.

Technical Summary

1. Problem Statement

In quantum computing platforms (particularly superconducting qubits), State-Preparation and Measurement (SPAM) errors often dominate over single-qubit gate errors. Accurate characterization of these errors is critical for:

• Developing effective error mitigation strategies.
• Improving hardware fidelity.
• Ensuring the reliability of quantum algorithms (e.g., phase estimation, error correction).

Limitations of Existing Protocols:

• Coupled Characterization: Standard protocols often estimate the product of SP and Measurement (M) errors ($\alpha_M \alpha_{SP}$) rather than separating them. This relies on the assumption that SP errors are negligible ($\alpha_{SP} \approx 1$), which is frequently invalid on current devices.
• Resource Intensity: Protocols that do separate these errors often require two-qubit gates (limiting accuracy to the fidelity of the two-qubit gate) or ancillary qubits, making them difficult to scale.
• Scalability: Some existing methods using single-qubit gates rely on circuits that scale polynomially with system size, allowing environmental noise to corrupt estimates.
• Bias in Mitigation: Current measurement-error mitigation techniques that ignore SP errors can lead to biased, and sometimes non-physical, estimates of observable expectation values.

2. Methodology: The QSPAM Protocol

The authors propose the Quantum SPAM (QSPAM) protocol, which characterizes SP and M errors independently using only high-fidelity single-qubit gates and repeated measurements without reset.

Core Assumptions

• High-fidelity single-qubit gates: Gate errors are negligible compared to SPAM errors.
• Non-destructive measurements: The qubit state survives the measurement process (or is reset to a known post-measurement state).
• Uncorrelated Errors: SP and M errors are spatially and temporally uncorrelated across qubits.
• Classical M Errors: Measurement errors are modeled as classical assignment errors (diagonal POVM elements), though the protocol can detect deviations from this.

Theoretical Framework

The protocol distinguishes between two versions:

1. sQSPAM (Simplified QSPAM): Assumes measurement (Kraus) operators are diagonal. It requires 5 experiments per qubit.
2. QSPAM (General QSPAM): Accounts for non-diagonal measurement operators (off-diagonal elements in the Kraus operators). It requires 8 experiments per qubit.

Key Mechanism:
The protocol utilizes repeated measurements without reset. By preparing a state, measuring it, and immediately measuring it again (conditioned on the first outcome), the protocol creates a system of nonlinear equations that decouples the initial state preparation fidelity ($\alpha_{SP}$) from the readout fidelity ($\alpha_M$).

• Standard SPAM: Measures $P_{z+ \to z-}$ and $P_{z- \to z+}$. This yields two equations with three unknowns ($\alpha_M, \delta, \alpha_{z,SP}$), requiring the assumption $\alpha_{z,SP} \approx 1$.
• QSPAM: Adds a sequence $P_{z+ \to z+ \to z+}$ (measure twice). This provides a third independent equation, allowing the unique solution of $\alpha_M, \delta,$ and $\alpha_{z,SP}$ without assuming perfect state preparation.
• Full Characterization: By rotating the initial state using Hadamard ($H$) and $\sqrt{X}$ ($SX$) gates before measurement, the protocol characterizes all three components of the SP Bloch vector ($\alpha_{x,SP}, \alpha_{y,SP}, \alpha_{z,SP}$).

Circuit Depth and Scalability

• The circuit depth is constant and independent of the system size ($N$).
• The protocol is highly parallelizable, allowing simultaneous characterization of all qubits in a device.
• Precision scales as $\nu^{-1/2}$ (where $\nu$ is the number of shots), limited only by statistical noise.

3. Key Contributions

1. Decoupling SP and M Errors: A rigorous method to extract SP and M parameters independently using only single-qubit operations, removing the need for the "perfect SP" assumption.
2. Efficiency: The protocol requires a fixed number of experiments (5 or 8) per qubit regardless of system size, avoiding the scaling issues of previous methods.
3. Detection of Non-Diagonal M Operators: The general QSPAM protocol can detect and quantify off-diagonal elements in measurement operators (parameterized by $\epsilon$), which standard diagonal models miss.
4. Bias Correction: Demonstrates that standard mitigation (ignoring SP errors) leads to overestimation of observable expectation values, particularly for collective observables like GHZ state stabilizers.

4. Experimental Results

The authors validated the protocol on IBM Quantum devices (specifically ibm_brisbane and ibm_sherbrooke).

• Validation of Independence: By artificially injecting SP errors (via $R_x(\phi)$ rotations) into the initialization, they showed that the characterized M-error parameter ($\hat{\alpha}_M$) remained stable while the SP parameter ($\hat{\alpha}_{z,SP}$) varied exactly as predicted. This confirmed the protocol successfully separates the two error sources.
• Device Characterization:
  • SP infidelities were found to be significant, reaching up to 6.57%.
  • Readout assignment errors reached up to 19.1%.
  • Most qubits exhibited diagonal measurement operators ($\epsilon \approx 0$), but Qubit 61 showed a significant non-diagonal component ($\hat{\epsilon} \approx 0.0015$), proving the necessity of the general QSPAM protocol.
• GHZ State Experiments:
  • The authors prepared GHZ states on up to 12 qubits and measured the collective observable $\langle Z^{\otimes N} \rangle$.
  • Standard Mitigation: Assumed perfect SP, leading to a significant overestimation of the expectation value (sometimes yielding non-physical values $>1$).
  • QSPAM Mitigation: Correctly accounted for SP errors, yielding accurate expectation values that matched theoretical predictions for noisy systems.
  • As the number of qubits increased, the difference between standard and QSPAM mitigation grew, highlighting the compounding effect of uncorrected SP errors.

5. Significance and Impact

• Hardware Benchmarking: Provides a resource-efficient tool for hardware developers to pinpoint whether performance bottlenecks lie in state preparation or readout, guiding targeted hardware improvements.
• Improved Algorithm Reliability: By enabling accurate SPAM mitigation, the protocol prevents the biasing of results in near-term quantum algorithms (NISQ), particularly those involving collective observables or error correction syndromes.
• Scalability: The constant-depth, parallelizable nature of QSPAM makes it suitable for large-scale quantum processors where two-qubit gate-based characterization is too noisy or slow.
• Theoretical Insight: The work highlights that ignoring SP errors in mitigation protocols is not just a minor approximation but a source of fundamental bias that can lead to non-physical conclusions.

In conclusion, the QSPAM protocol offers a practical, scalable, and theoretically rigorous solution to the long-standing challenge of separating state-preparation and measurement errors, significantly enhancing the reliability of quantum characterization and error mitigation on current and future quantum hardware.