⚛️ quantum physics

Testing time order and Leggett-Garg inequalities with noninvasive measurements on public quantum computers

This paper demonstrates the first violation of Leggett-Garg inequalities and time-order noninvariance on public quantum computers using genuine noninvasive measurements, leveraging new fractional gates to establish weak measurement protocols as sensitive benchmarks that reveal statistically significant deviations from theoretical predictions beyond declared device error rates.

Original authors: Tomasz Rybotycki, Tomasz Białecki, Josep Batle, Bartłomiej Zglinicki, Adam Szereszewski, Wolfgang Belzig, Adam Bednorz

Published 2026-01-29

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Tomasz Rybotycki, Tomasz Białecki, Josep Batle, Bartłomiej Zglinicki, Adam Szereszewski, Wolfgang Belzig, Adam Bednorz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Peeking Without Touching

Imagine you have a very delicate, spinning top. In the classical world, if you want to know which way it's spinning, you can just look at it. Your eyes don't really change how it spins.

But in the quantum world (the world of tiny particles like atoms), looking at something is a violent act. If you try to "look" at a quantum particle to see its state, you usually knock it over or change its direction. This is called a projective measurement. It's like trying to check the temperature of a cup of coffee by sticking a giant, freezing thermometer into it; the act of measuring changes the coffee.

For a long time, scientists wanted to test a specific rule called the Leggett-Garg Inequality. This rule asks: Does a quantum object have a definite state even when we aren't looking at it? (This is the concept of "realism"). To test this, you need to peek at the object at different times without changing its path. But because normal "peeking" changes the object, the test was impossible to do cleanly.

The Solution: The "Ghostly" Touch

This paper describes a team that finally did this test on public quantum computers (like the ones you can rent online from IBM and IonQ). They used a technique called Weak Measurement.

Think of a weak measurement like a ghostly touch.

Normal Measurement: A strong punch that knocks the top over.
Weak Measurement: A gentle breeze that barely nudges the top. It's so light that the top keeps spinning mostly the same way, but the breeze carries a tiny bit of information about the spin.

The catch? A single breeze is too faint to tell you much. You need to feel the breeze thousands of times and average the results to get a clear picture. The team did exactly this, gathering massive amounts of data to see the "ghostly" pattern.

The Experiment: The Time Travel Test

The researchers set up a game with three steps involving a quantum particle and two "ghostly" sensors (let's call them Sensor A and Sensor B), followed by a final check (Sensor C).

The Setup: They prepared a quantum particle in a specific state.
The Test: They measured the particle with Sensor A, then Sensor B, then Sensor C.
The Twist: They also ran the experiment in reverse order: Sensor B, then Sensor A, then Sensor C.

In our everyday world, the order of events shouldn't matter if you are just gently observing. If you check the weather in the morning and then the afternoon, it shouldn't matter if you say "Morning then Afternoon" or "Afternoon then Morning"—the data should be the same. This is called Time-Order Invariance.

What They Found

The results were shocking and confirmed that the quantum world is very strange:

Breaking the Rules (Leggett-Garg Violation): The data showed that the particle did not have a definite state before they looked at it. The "ghostly" measurements revealed that the particle's reality was created by the act of measuring it. They violated the Leggett-Garg inequality by a huge margin (more than 5 to 10 times the expected error rate).
Order Matters (Time-Order Violation): When they swapped the order of the sensors (A then B vs. B then A), the results were completely different. In the quantum world, the sequence of "gentle touches" changes the outcome. It's as if checking the weather in the afternoon before the morning actually changed the morning's weather.

The Hardware: Public Computers

The team didn't build a special lab machine. They used public quantum computers available on the internet (IBM and IonQ).

They tested on 10 different groups of 3-qubit "circuits" across 5 different devices.
They used new, specialized "fractional gates" (which are like dimmer switches for quantum operations) to create these gentle, weak measurements.
They found that while the computers were noisy (like a room with a lot of background chatter), the signal was so strong that they could still clearly see the quantum weirdness.

The Conclusion

The paper claims that they have successfully used public quantum computers to prove two things:

Quantum objects do not have a fixed reality until they are measured (violating "macrorealism").
The order in which you gently measure a quantum system matters (violating "time-order invariance").

They did this without "freezing" the system or destroying it with strong measurements, proving that these public machines are now powerful enough to test the deepest, most philosophical questions about how reality works.

Technical Summary: Testing Time Order and Leggett-Garg Inequalities with Noninvasive Measurements on Public Quantum Computers

Problem Statement
Quantum mechanics challenges classical intuitions regarding realism and temporal order, most notably through the violation of local realism and the Leggett-Garg (LG) inequality. The LG inequality, derived under the assumptions of macrorealism and noninvasiveness, posits that measurement outcomes should not depend on the order of observation if the system is not disturbed. However, standard projective measurements are inherently invasive, often freezing system dynamics (quantum Zeno effect) or disturbing subsequent evolution, thereby preventing a genuine test of noninvasiveness. While weak measurements offer a theoretical solution by minimizing disturbance (scaling with the square of the measurement strength), previous experimental realizations on quantum hardware have been limited. They often relied on post-selection, frequency-domain mapping, or strong entangling operations that did not fully decouple the measurement strength from the disturbance. Furthermore, previous tests on public quantum computers were restricted to projective measurements, which cannot be considered noninvasive in the quantum sense.

Methodology
The authors implemented a protocol to test LG inequalities and time-order invariance using genuine weak measurements on public quantum devices from IBM Quantum (Eagle and Heron generations) and IonQ.

Weak Measurement Scheme: The experiment utilizes a Positive Operator-Valued Measure (POVM) approach where the system interacts with an auxiliary "meter" qubit. The interaction strength is controlled by a parameter $\lambda$ (or angle $\theta$ ), such that the disturbance scales as $\lambda^2$ while the information gain scales as $\lambda$ .
Protocols: Two distinct protocols were implemented to realize weak measurements:
1. Protocol I: Relies on a single entangling operation (e.g., CX, CZ, or ECR) combined with a specifically prepared meter state ( $|\Omega_\theta\rangle$ ) close to an eigenstate of the operation. This minimizes invasiveness by ensuring the operation leaves the system unchanged if the meter is in the ideal state.
2. Protocol II: Utilizes two self-reversing entangling operations (or a single fractional gate) to create an operation close to the identity ( $U \approx 1$ ). This approach is independent of the initial meter state.
Hardware Implementation:
- IBM Devices: The authors tested 10 sets of 3-qubit groups across five devices (ibm_brisbane, ibm_sherbrooke, ibm_kyiv, ibm_torino, ibm_kingston). They employed native gates including ECR (Echoed Cross-Resonance) and, crucially, new fractional gates ( $X_\theta$ , $RZZ_\theta$ ) available on Heron devices to fine-tune measurement strength.
- IonQ Device: A single 3-qubit group on the Forte Enterprise 1 device was tested using native all-to-all connectivity and fractional $RZZ$ gates.
Experimental Design: The test involves three observables ( $A, B, C$ ). $A$ and $B$ are measured weakly, while $C$ is measured projectively. The experiment compares two time orders: $A \to B \to C$ and $B \to A \to C$ . To mitigate drift and systematic errors, the authors collected large statistics (up to 100,000 shots on IonQ) and calculated the difference between results for positive and negative measurement strengths ( $\theta = \pm 0.1$ ).

Key Contributions

First Genuine Noninvasive Violation on Public Hardware: This work demonstrates the first violation of LG inequalities and time-order invariance on public quantum computers using genuine noninvasive (weak) measurements, avoiding the need for post-selection.
Utilization of Fractional Gates: The authors benchmarked newly available fractional gates on IBM Heron devices, showing their applicability for weak measurement protocols.
Statistical Rigor: By gathering sufficiently large statistics, the authors achieved violations exceeding 5 standard deviations in almost all tested qubit sets, with many reaching 10 standard deviations.
Hardware Benchmarking: The study establishes weak measurement protocols as a sensitive benchmark for quantum hardware. The data revealed statistically significant deviations from theoretical predictions that exceeded the declared device error rates, suggesting that weak measurements can detect subtle hardware imperfections.

Results

Leggett-Garg Violation: The experiments confirmed the violation of the LG inequality ( $\langle A \rangle + \langle B \rangle - \langle AB \rangle \leq 1$ ) on most qubit groups. The observed values exceeded the classical limit of 1, approaching the quantum theoretical maximum of $\sqrt{2}$ .
Time-Order Violation: The experiments demonstrated that quantum correlations depend on the order of measurement ( $\langle ABC \rangle \neq \langle BAC \rangle$ ), violating the classical assumption of time-order invariance. The difference $\langle ABC \rangle - \langle BAC \rangle$ was observed to be non-zero, consistent with the theoretical prediction of $1/2$ .
Device Performance:
- IBM Eagle: Showed consistent violations across most qubit groups using ECR and CZ gates.
- IBM Heron: Results varied; while time-order violations were consistent, LG violations were more sporadic (e.g., only one group on torino passed, while at least five on kingston did).
- IonQ: Showed violations of both LG and time-order invariance, with some results even exceeding theoretical quantum expectations. The authors attribute this excess to potential miscalibration of the fractional gate angles, noting that a ~5% deviation could explain the discrepancy.
Noninvasiveness Verification: The authors verified that the measurement was noninvasive by showing that the correlations differed significantly from those obtained via projective measurements. The data agreed qualitatively and quantitatively with the model of weak disturbance.

Significance and Claims
The paper claims that these advances transform public quantum computers into practical testbeds for probing foundational questions of realism and temporal order with unprecedented accessibility and precision. By successfully implementing weak measurements without post-selection, the authors demonstrate that public quantum hardware can be used to test the limits of classical realism.

The authors remain modest regarding the interpretation of the hardware deviations. They acknowledge that while the violations are significant, the large deviations from theoretical predictions (exceeding declared error rates) require urgent explanation and suggest that weak measurement protocols serve as a highly sensitive benchmark for quantum hardware calibration. They do not claim to have solved the issue of invasiveness entirely but emphasize that the "limiting behavior" of their weak measurements justifies the claim of noninvasiveness within the context of the experiment. The work highlights both the progress in quantum hardware (e.g., IonQ and IBM Heron) and the remaining challenges in gate calibration and error mitigation.