Our Novel, Efficient Approach to Quantum Error Correction

An Error Correction Code Breakthrough at IonQ

IonQ researchers recently published details on a novel approach to quantum error correction. Achieving fault-tolerant error correction is widely recognized as a long term goal for the quantum computing community. One of the main challenges to error correction schemes is they are resource-intensive, requiring a large number of physical qubits per logical qubit, making them impractical for near-term quantum computers. But IonQ has just made huge progress in making the journey to error correction more seamless.

IonQ's new method, dubbed CliNR (pronounced 'cleaner'), short for Clifford Noise Reduction, is an efficient and scalable approach that can improve performance for the noisiest subset of quantum gates on today's quantum computers. Unlike traditional quantum error correction (QEC) that requires significant overhead or quantum error mitigation (QEM) which faces exponential sampling overheads, CliNR is an efficient addition to error correction schemes because it provides lower overhead. With a qubit overhead of ~3:1 and a gate overhead of 2:1, CliNR has the potential to be a 'partial error correction' approach, filling the gap between today's quantum error mitigation approaches and the long term promise of fully error corrected, fault-tolerant quantum computing.

IonQ's novel CliNR method is capable of reducing the logical error rate (the probability that an error occurs in the logical qubit of a quantum computation, despite the use of error correction techniques) by a factor of two.

Based on modeled assumptions of native gate fidelities, CliNR is able to reduce the probability of an error within circuits, scaling efficiently with circuit size and depth.

Innovating To Address the NearTerm Error Handling Value Gap

QEM and QEC are two leading methods to deal with errors in quantum computation. To compare these techniques, we consider the resources they consume in terms of:

• number of qubits

• number of quantum gates

• number of shots

where the number of shots refers to the number of executions of a quantum circuit.

Today's quantum computing users are facing a value gap, or a gap between the Quantum Error Mitigation (QEM) schemes adding value today but approaching their scaling limits, versus Quantum Error Correction (QEC) schemes that are scalable in the long term but that remain prohibitively expensive today.

The following diagram summarizes the error handling value gap and associated short-term versus long-term costs.

Our new work ¹ fills this gap with a new method for partial error correction with a small qubit cost, a small gate cost, and a small shot cost. By reducing the noise of a broad class of quantum circuits called Clifford circuits, our CliNR method unlocks a whole new 'partial error correction' approach to solving problems.

Our method is not designed to replace QEC in the long term, as it does not offer the same level of protection as QEC, but it does present a low-overhead alternative to QEC in the near term. That said, there is still applicability of the approach in the long term. For example, CliNR could also be used on logical qubits - in combination with QEC - lessening the requirement for QEC, thereby further reducing overhead costs.

QEM is already used to great effect in today's IonQ machines: Harmony, Aria, and Forte all leverage QEM techniques. While QEM uses a relatively small number of qubits and gates, it is expected to scale poorly as qubit counts continue to grow and shot costs increase. As circuit size grows, QEM either ceases to be effective or it incurs a runtime overhead that grows exponentially. A different solution is needed for the long term.

This is where QEC comes into play because it does not suffer from this exponential increase in shot count. However, the cost of QEC in terms of qubit count and gate count is still very significant, and currently out of reach for practical use. For example, to execute a large-scale quantum algorithm with the surface code, one may need thousands of physical qubits per logical qubit. Recent breakthroughs based on quantum low density parity-check (qLDPC) codes reduce the qubit cost of QEC, but they still consume more than 20 physical qubits per logical qubit ²³⁴⁵.

CliNR acts as middle ground and solves for an important pain point, offering an efficient approach for reducing logical errors in Clifford circuits with lower overhead than QEC and better scalability than QEM, and it should work in the near term.

Applying CliNR to Unlock Customer Value

We expect that CliNR can be applied in multiple ways. First, customers will use it as a means of improving near term algorithmic results' accuracy. In today's IonQ systems, two-qubit gates are where most of the noise is introduced. This is why our original motivation was to focus on the correction of two-qubit gates. The CliNR method we designed is capable of correcting all Clifford gates, which is a broad class of quantum gates that includes common two-qubit gates like CZ, CNOT, as well as maximally-entangling XX and ZZ native to IonQ devices. Since this error compounds across the depth of a circuit, applying CliNR has the potential to enable deeper circuits and more accurate results.

Second, the low overhead of CliNR makes it an attractive option for reducing Clifford gate noise prior to being applied in broader error correction schemes that require higher input gate fidelities to achieve target logical error rates. These are just two key examples we expect to see play out in the real-world application of CliNR.

Unpacking the Innovation Behind CliNR

Error correction is not solely a quantum concept. To describe the general idea of CliNR, we can consider a similar problem in the classical world. Imagine that we want to compute the sum of 1,000 integers using a noisy adder with a 1% noise rate. This noisy adder computes the sum s = x+y of two numbers x and y; however, 1% of the time, instead of s it returns s+e where e is a random number between 0 and s. Let's see what happens when we compute the sum 1 + 2 + 3 + … + 1,000 with this noisy adder. The result should be 500,500. We simulated this computation with a noisy adder and ran the computation 20 times. Here are the results.

None of them are correct! This is because the noise rate (1%) is too high to add 1,000 numbers reliably. Indeed, there are on average 10 errors during this computation.

The computation of shorter sums is less noisy. For example, the same simulation for the sum of the first 100 integers, gives the following results:

The correct result (5,050) appears six times. Therefore, we can identify the correct answer by a majority vote.

To make a sum of many numbers less noisy, we split the sum into sub-sums. Each sub-sum is computed twice and, if the two results disagree, we restart the computations of this sub-sum. The result is obtained by adding all the sub-sums.

This time, only three of the 20 simulations give an incorrect sum, whereas the 20 previous results were all incorrect without the sub-sum verification. This approach significantly reduces the noise rate of the computation.

To recap, the key features of this strategy are:

• Break the overall problem down into smaller pieces, each of which is less likely to be faulty.

• Keep the number of integers inside each sub-sum small enough to avoid a spike in the number of restarts.

• Keep the number of sub-sums small enough to make the result reliable.

The CliNR method described in our paper can be seen as a quantum variant of this strategy. Below, we show a quantum circuit implemented using CliNR. The resemblance with the classical adder example is evident in the design.

The CliNR circuit repeats with the same verification logic on sub-components of the circuit to reduce error probability, analogous to the classical adder example above.

When moving to the quantum setting, the main challenge is that one cannot readout the state of a set of qubits without potentially disturbing that state. Therefore, one must use a different strategy to verify intermediate results - by exploiting the structure of Clifford circuits, together with the gate teleportation scheme. We invite readers to have a look at our technical paper for more details ⁶.

Industry Progress and The Path Forward

It is an exciting time for the quantum computing community. Several building blocks of a fault-tolerant quantum computing architecture have been demonstrated experimentally, such as the Bacon-Shor code ⁷, the five-qubit code ⁸⁹, the Steane code ¹⁰, the surface code ¹¹¹²¹³, logical IQP circuits ¹⁴, fault-tolerant teleportation ¹⁵, and a logical quantum Fourier transform ¹⁶. These experiments provide critical insights into the long-term goal of building a large-scale fault-tolerant quantum computer.

Despite the material progress being made on QEC, quantum computing hardware must still improve significantly both in scale and quality before useful quantum error correction can be realized. While we work hard to solve the hardware challenge, we've also created a novel way - the CliNR method - to derive maximum quantum value in the near-term. CliNR is a practical and scalable alternative to conventional quantum error correction in the near-term, providing significant error reduction with minimal resource overhead, making it a promising technique for practical quantum computing. Further, we expect this to be a robust and enduring approach that will pay dividends in the long-term, as it will maintain relevance even in the era of fault-tolerant error correction.

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