Quantum error correction represents a cornerstone for developing large-scale quantum computers. By integrating numerous physical qubits—our superconducting circuits storing quantum information but susceptible to noise—we create robust logical qubits resistant to errors. Recently, we achieved a significant milestone: implementing a surface code quantum error correction experiment demonstrating performance exceeding the threshold required for scaling benefits. This advancement theoretically enables the creation of near-perfect logical qubits simply by increasing physical qubit count.
Key future challenges involve minimizing physical qubits per logical qubit and accelerating logical operation and quantum algorithm speeds. While enhancing physical qubit quality (reducing error rates) directly addresses these, improving error correction code efficiency offers another powerful path. This leads us to our exciting experimental demonstration of a “color code” system, presenting an efficient alternative to the widely studied surface code. Our recent publication in Nature, “Scaling and logic in the color code on a superconducting quantum processor,” details the essential building blocks for a resource-efficient, fault-tolerant quantum computer utilizing the color code. Like the surface code, the color code encodes logical qubits using multiple physical qubits to detect and correct errors in real-time. However, its distinct triangular pattern of parity measurements, using hexagonal tiles, requires fewer physical qubits and enables more efficient logical gates than the surface code, though it necessitates deeper physical circuits and a different decoding algorithm.
Triangles Outperform Squares in Quantum Error Correction Efficiency
Evaluating error correction codes involves assessing their “distance,” which quantifies the number of physical errors leading to a single logical error. The surface code requires a square layout where the edge length equals the code distance. In contrast, the color code utilizes a hexagonal tiling forming a triangular shape, also with an edge length corresponding to the code distance. This triangular geometry significantly reduces the color code’s area compared to a surface code of equivalent distance, indicating a reduced need for physical qubits.
Comparing the triangular color code to the square surface code. The geometry of the color code is advantageous because it uses fewer physical qubits for the same code distance.
However, this efficiency comes with a tradeoff. Implementing the error correction circuit and decoding results proves more complex in the color code, potentially impacting the accuracy of matching algorithms. Consequently, reaching the error correction threshold—the performance level necessary for effective error correction—becomes more challenging. Despite this, our latest publication demonstrates that our Willow chip, combined with recent decoding advancements, achieves below-threshold performance with the color code. Our experiments comparing color codes with distances of 3 and 5 revealed a logical error rate suppression of 1.56× at the higher distance. While this figure is less than the 2.31× suppression previously achieved with surface codes, we anticipate the color code’s geometrical advantage will become more pronounced at larger scales and with further hardware improvements.
Data showing the rate of error accumulation in the color code. In this plot, we see that the distance-5 color code outperforms the distance-3 color code (by accumulating errors more slowly). Showing that the higher distance color code would perform better, means that we can, in principle, obtain much higher logical performance by simply adding more qubits.
Accelerated Single-Qubit Logical Operations with Color Codes
The color code demonstrates significant advantages in logical operations, where certain single-qubit operations become notably simpler than in the surface code. Many color code logical operations can execute in a single step, contrasting with the surface code’s requirement for numerous error correction cycles to achieve the same outcome. For instance, a logical Hadamard gate is expected to execute in approximately 20ns within the color code, a task that could take 1000 times longer using a surface code on the same device. This speed enhancement dramatically accelerates logical gate execution and, consequently, the performance of quantum algorithms. Furthermore, requiring fewer error correction cycles per algorithm allows for greater error tolerance per cycle, thus reducing the need for physical qubits.
Leveraging this capability, we successfully performed numerous single-qubit logical operations and verified the outcomes through “logical randomized benchmarking,” detailed further in our paper.
Efficient Injection of Arbitrary States for Quantum Computing
A critical component of quantum computing is the generation of “magic states,” also known as T-states, which are essential for creating arbitrary qubit rotations. Without these rotations, quantum algorithms cannot surpass classical computational methods. Historically, magic state generation was considered the most resource-intensive aspect of practical quantum algorithms. However, recent theoretical breakthroughs demonstrate that the color code enables an efficient circuit, termed “cultivation,” for generating these crucial magic states. This innovation significantly boosts the potential of future quantum algorithms.
This finding underscores the importance of implementing color codes in our quantum systems. Our experiment successfully executed the initial phase of the “cultivation” protocol, injecting an imperfect magic state into a color code logical qubit with 99% fidelity.
Stitching Patches for Advanced 2-Qubit Operations
Another essential operation for running quantum algorithms involves 2-qubit gates. These are typically implemented by merging two distinct logical qubits into a single patch, then separating them. In our experiments, we effectively demonstrated this technique by entangling two logical qubits and transferring information between them with fidelities ranging from 86% to 91%. An interesting aspect of the color code is its enhanced flexibility, allowing merge operations across three bases (X, Y, and Z), compared to the two available in the surface code (X and Z).
Diagram showing the merging and separation of two logical qubits to transfer the logical state (represented by the bolded black line) from one patch to the other.
The Future of Quantum Computing: Color Codes and Beyond
After years of prioritizing the surface code, our latest findings reveal compelling alternatives capable of efficient implementation below their error correction threshold on superconducting quantum processors. While the surface code remains our primary configuration, the color code is poised to become an integral part of our large-scale quantum computer. Its crucial role in the “cultivation” protocol promises more efficient magic state injection. Furthermore, as our quantum hardware advances, the color code may ultimately surpass the surface code in efficiency, both in terms of physical qubit requirements and error correction cycles.