Encoding and Syndrome Measurement in Logical Qubits

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome back to our analysis of landmark quantum error correction experiments.

In the previous lesson, we established the concept of the "break-even" point and surveyed a few recent breakthrough results. We identified the GKP code experiment by Sivak et al. as a prime candidate for a deep dive.

To ensure you get a broad perspective on the different experimental paradigms being pursued, we will pivot our focus in this lesson. Instead of the continuous-variable GKP code, we will analyze a different, but equally significant, recent paper that demonstrates a qubit-based quantum low-density parity-check (qLDPC) code. As you'll recall from our earlier lessons, qLDPC codes are theoretically very promising, but their non-local connections pose a major experimental challenge. Analyzing how a leading research group tackled this challenge will provide valuable insights into the state of the art.

Today, we will address the learning outcome: Analyze the experimental methodology for encoding the logical qubit and performing syndrome measurements as described in the selected paper.

Our selected paper for this lesson is a 2025 preprint from a team including researchers from the University of Science and Technology of China: "Demonstration of low-overhead quantum error correction with a bivariate bicycle code".

Our goals are to dissect the how of this experiment by exploring:

The specialized hardware designed to support a qLDPC code.
The method used to encode logical qubits into the physical system.
The circuit and procedure for performing the crucial syndrome measurements.

1. The Challenge of qLDPC Codes and the Hardware Solution

As we've discussed, the power of codes like the surface code comes from their geometric locality: stabilizers only involve physically adjacent qubits. This makes them a natural fit for 2D quantum processors. In contrast, qLDPC codes sacrifice this locality to achieve better encoding rates (more logical qubits for a given number of physical qubits). Their stabilizers often connect distant qubits.

This immediately raises the central experimental question: How do you build a processor that supports these non-local interactions? The paper we are studying provides a direct answer.

Demonstration of low-overhead quantum error correction ...

Let's begin by examining the custom quantum processor, 'Kunlun', that was designed for this experiment. Please read the sections describing the experimental setup and the device details from the main text and supplementary information.

First, read the section 'Experimental setup and framework' in the main text. Then, for more detail, read 'II.1 Device Information' in the supplementary material. Focus on identifying the key architectural features that distinguish this processor from a standard nearest-neighbor-coupled chip and enable the implementation of a qLDPC code.

The reading highlights two critical hardware innovations that make this experiment possible:

Long-Range Tunable Couplers: Instead of only connecting adjacent qubits, the Kunlun processor features couplers of varying lengths (from ~0.3 mm to ~2.6 mm). Each check qubit is physically connected to six data qubits, some of which are several lattice sites away. This directly implements the non-local connectivity required by the chosen qLDPC code's Tanner graph (shown in their Fig. 1a).
Air-Bridges: To physically realize this high-connectivity graph on a 2D chip without the couplers interfering with each other, the engineers built a quasi-3D structure. Air-bridges allow some couplers and control lines to "fly over" others, solving the complex wiring crossover problem.

Your physics background gives you an appreciation for the complexity of superconducting circuits. These engineering feats—realizing tunable couplers of different lengths with consistent performance and managing the intricate layout with air-bridges—represent a significant step beyond the simpler, grid-like architectures used for surface codes.

2. Encoding the Logical State: A Projective Approach

Now that we understand the hardware, let's look at how the logical information is initialized. The experiment implements a specific type of qLDPC code, a bivariate bicycle (BB) code with parameters [[16, 4, 4]]. This notation, [[n, k, d]], means it encodes k=4 logical qubits into n=16 physical data qubits, with a code distance of d=4.

One might imagine a complex circuit that takes 4 qubit inputs and entangles them across 16 data qubits. However, the experimental procedure is more subtle and common in QEC experiments.

Demonstration of low-overhead quantum error correction ...

The paper describes how the logical basis states are prepared. This is not done via a deterministic encoding circuit but through a projective method. Please read the relevant passages from the main text and the supplementary information that detail this process.

First, read the paragraph in the main text under the heading 'With all elements in place...'. This gives the high-level summary. Then, for the mathematical foundation, read the first few paragraphs of 'I.1 Bivariate bicycle code construction' in the supplementary information, down to the definition of the parity-check matrix for a CSS code. This will clarify the roles of Z-type and X-type stabilizers.

The encoding methodology is as follows:

Random Initialization: The 16 data qubits are not prepared in a specific state. They are simply initialized randomly into either $∣0 ⟩$ or $∣1 ⟩$ . This state is, of course, not in the code space.
Projective Measurement: A full cycle of stabilizer measurements is performed. As you know from the stabilizer formalism, measuring the stabilizers of a code projects the system's state into the codespace (or more precisely, a specific eigenstate of the measured logical operators).
Defining the Initial State: The outcomes of this first measurement round define the initial logical state. For example, the paper states that the logical state $∣ L_{i} ⟩$ of the $i$ -th logical qubit is determined by the measured parity of its corresponding logical $Z$ operator. If the parity is even, the logical qubit is defined as being in the $∣ 0_{L} ⟩$ state; if odd, it's in $∣ 1_{L} ⟩$ .

This projective encoding is elegant because it bypasses the need for a potentially complex and error-prone state preparation circuit. The first syndrome measurement round serves a dual purpose: it prepares the logical state and provides the initial reference against which future errors will be detected.

3. Simultaneous, Non-Local Syndrome Measurement

The core of any active QEC scheme is the repeated measurement of error syndromes. For this BB code, this means measuring the 14 independent stabilizers, each of which has a weight of 6 (i.e., it acts on 6 data qubits). Doing this efficiently and accurately is the central challenge.

The authors implement a circuit that measures all stabilizers simultaneously. This is a major departure from early surface code experiments that often measured stabilizers sequentially.

Demonstration of low-overhead quantum error correction ...

Let's examine the heart of the experimental procedure: the syndrome measurement circuit. The authors use a highly optimized circuit to measure all weight-6 stabilizers at once. Please read the sections describing this circuit.

Start with 'I.2 Syndrome measurement circuit' in the supplementary info for the high-level design. Then, look at Figure 2c in the main text and 'II.3 Characterization of stabilizer measurements' in the supplement to see the circuit for a single stabilizer. Finally, read 'With high-precision, long-range two-qubit gates...' in the main text to see how individual measurements are integrated into a full cycle.

The key aspects of their syndrome measurement methodology are:

The Basic Building Block: The circuit for measuring a single weight-6 stabilizer (e.g., a Z-type stabilizer, $S_{i}^{Z}$ ) is shown in Figure 2c. It uses an ancilla (check) qubit, initially prepared in $∣0 ⟩$ , and a sequence of CNOT gates controlled by the 6 data qubits. For a Z-stabilizer, this is preceded and followed by Hadamard gates on the ancilla.
Simultaneous Execution: The full syndrome extraction circuit, shown in Fig. S3, is composed of seven layers of non-overlapping CZ gates. This is a critical result. It means that despite the code's non-local structure, all stabilizers can be measured in a constant-depth circuit. The number of layers doesn't grow as the code size increases, which is a vital property for scalability.
Error Mitigation: The authors employ several techniques to improve accuracy. For instance, as seen in Fig. S7b, they insert Pauli-X gates between adjacent CZ gates. This is a form of dynamical decoupling that mitigates dephasing errors on the qubits while the gates are being executed.
From Readout to Syndrome: After measuring the check qubits, the raw outcomes must be converted into error detection events. As described in the supplement (Section III.1), because the check qubits are not reset between cycles, the syndrome for cycle t is calculated by taking the XOR of the measurement outcomes from cycle t and cycle t-1. An "error detection event" is then recorded if the syndrome in the current cycle differs from the previous one.

Test your understanding!

Consider the circuit for measuring a single Z-type stabilizer involving 6 data qubits $D_{1}, ..., D_{6}$ and one ancilla qubit $A$ . The circuit is: H(A) -> CNOT(D1, A) -> ... -> CNOT(D6, A) -> H(A) -> Measure(A).

If the state of the data qubits has even parity (an even number of qubits are in the $∣1 ⟩$ state), what is the final state of the ancilla qubit before the measurement, and why?

Show answer

The ancilla starts in $∣0 ⟩$ . The first Hadamard gate puts it into the $∣ + ⟩ = \frac{1}{2} (∣0 ⟩ + ∣1 ⟩)$ state.
Each CNOT gate flips the ancilla qubit if its control data qubit is in the $∣1 ⟩$ state. An even number of flips on the $∣1 ⟩$ component of the ancilla's superposition state results in no net change (or a phase of $(- 1)^{even} = + 1$ ).
Therefore, after all 6 CNOTs, the ancilla remains in the $∣ + ⟩$ state.
The final Hadamard gate transforms $∣ + ⟩$ back to $∣0 ⟩$ .
Thus, a measurement will yield the outcome '0', correctly indicating even parity. Conversely, odd parity would result in the ancilla being flipped to $∣ - ⟩$ and then measured as '1'.

Conclusion

In this lesson, we have dissected the experimental methodology of a cutting-edge qLDPC code demonstration. By moving beyond the nearest-neighbor paradigm, this work showcases a viable path for implementing codes with potentially superior properties to the surface code.

Key Takeaways:

Hardware for Non-Locality: The Kunlun processor's use of long-range tunable couplers and air-bridges is a direct and effective solution to the connectivity challenge posed by qLDPC codes.
Projective Encoding: The logical qubits are prepared not by a dedicated circuit, but by projectively measuring the stabilizers of an initially random state. The results of this first measurement define the initial logical information.
Efficient Syndrome Measurement: All non-local, weight-6 stabilizers are measured simultaneously using a constant-depth circuit of only seven CZ gate layers, demonstrating a scalable approach to syndrome extraction.

Preview of the next lesson:
We now have a clear picture of how the experiment was performed and how the raw data (the syndrome measurements over time) was generated. The next logical question is: How do you use this data to determine if the experiment was a success? In our final lesson of this project, we will analyze the data analysis techniques used to calculate logical error rates and critically assess the evidence presented for achieving error correction. We will see how the stream of syndrome bits is fed into a decoder to infer the most likely errors and evaluate the performance of the logical qubit.