A Hardware-Efficient Approach to Quantum Error Correction

Quantum computing is a nascent, yet incredibly promising field which has the potential to unlock important innovations in a range of industries from finance to pharmaceuticals. And while the industry is getting closer to achieving this with each passing week and month, there remains some difficult challenges which must be overcome before the full potential of quantum computers is unleashed. 

There is some degree of consensus in the industry that in state-of-the-art quantum computing, no matter the physical implementation of the system in question (superconducting, trapped ion, neutral atom and so on), at present there are simply far too many errors in all of these systems, and too often, to perform useful computation. 

How then can we proceed towards the promised land known as fault- tolerant quantum computing?

Many believe that our best hope is to implement quantum error correction (QEC) and reduce the amount of errors in these systems to a more manageable level. However, the hardware requirements to sufficiently correct errors are widely recognized as being extremely daunting. In most cases, 1,000 physical qubits or more are required as a backup for each logical qubit. Meaning that when the logical qubits are performing a calculation and an error occurs, one of the physical qubits is called upon to take over before the quantum information is lost due to the error.

If that sounds strange to you, you’re not alone. This is the exact challenge Nord Quantique was founded to solve.

The idea is to develop an approach that relaxes these stringent requirements by implementing QEC directly at the hardware level. Based on superconducting circuits, this approach maintains fast clock speeds in the megahertz range, while promising a shorter path to fault-tolerant computation than other competing approaches. 

The catalyst for this approach is designing error correction built-into the system at the hardware level.

To see how this form of error correction can be implemented at the hardware level, it is helpful to examine the guiding principle of QEC. Much like its classical counterpart, QEC works through understanding noise in the environment surrounding these systems, which causes disruptions to the quantum state of the machine. Once that is achieved, teams can then design codes and control processes that manipulate the system in ways the environment cannot. For instance, a classical error correction code for transmitting a message to the moon is designed differently than the code used to secure information on a hard drive. This code, when correctly implemented, protects important logical information from being lost, so long as sufficiently few errors have occurred.

The standard approach to QEC in superconducting circuits relies on locality in physical space by placing many qubits organized in a lattice across a processor chip. The noise that causes energy loss is not correlated over long distances. Therefore, qubits that are sufficiently far apart will suffer independent errors. On the other hand, quantum processors are designed with control signals that may manipulate information in a correlated fashion across the entire processor chip. If the logical information is then stored nonlocally across the chip, then the controller has the chance to manipulate the information and correct the local errors before they spread to corrupt the logical information. Furthermore, if error correction is applied quickly and constantly, the calculations being performed are much more likely to succeed.

While this approach to QEC is being pursued by many academic groups and quantum computing companies alike, performing this type of error correction also causes errors in and of itself. The quality of control operations and the sheer number of qubits required to correct more errors than these operations generate is simply not available with current hardware. And there is still a long way to go in this area. 

It was clear to a small group of Canadian researchers at the University of Sherbrooke (and others, like myself while at Yale University) that a different approach would be required. Enter Nord Quantique’s approach to developing qubits with built-in error correction, leveraging a nearly harmonic oscillator.

But how does a single harmonic oscillator have built-in error correction? These are designed and implemented based on the same QEC principles as outlined above. Except, in this case, relying on locality in the phase space of a single oscillator instead of in physical space, which is intrinsically space efficient. A mechanical analogy for the electromagnetic oscillator here is helpful: picture a car driving along a racetrack. The first step of designing a QEC code is to know the noise. For the car, the noise might be in the form of an uneven road, slowly changing the direction or speed of the vehicle as it passes over bumps. We say this noise is local because it is highly unlikely the car will teleport suddenly to another location in space, instead it will be perturbed locally first. Said another way, if the car hits a bump on the track, it will likely change trajectory slightly. However, not to the degree that one would be unable to predict where the car will be at a later time, and with only a little uncertainty proportional to the size of the bump. If the bump was sufficiently small, you can then easily correct it and set the car back on course with a minor correction of the steering wheel. Noise in electromagnetic oscillators is similarly local; it adds a ‘fuzz’ of uncertainty on the magnitude and direction of an electric field without changing it completely.

Nord Quantique’s QEC code of choice is based on grid states in a harmonic oscillator. It was originally designed by Gottesman, Kitaev, and Preskill (GKP) to correct this added fuzz, and has since been shown to be optimal for correcting energy loss. Because the grid in these GKP states extends far in phase space, also making them non-local, the information can therefore be stored nonlocally while the noise acts locally to add fuzz. Specifically, the logical information is stored in the phase of the superposition between distant pieces of the grid, allowing the key data to be preserved at a safe distance. Thus, if the local fuzz can be efficiently corrected and reduced before it spreads sufficiently far, the nonlocally stored logical information remains protected from it.

To perform this correction and manipulate the logical information, we require a method of implementing nonlocal control on a harmonic oscillator. For this, Nord Quantique utilizes an auxiliary control qubit. By implementing special entangling gates between the oscillator and the auxiliary qubit, we map the noisy fuzz information away from the oscillator to the auxiliary and evacuate the noise away from the system by resetting the auxiliary. This keeps the oscillator in a robust condition to deal with future errors. In effect, it implements a modular position measurement that gives access to the noise without spoiling the underlying logical information.  And, critically for the performance of these machines, all this can be done with clock speeds in the MHz range which superconducting architectures are known for.

Going back to the car analogy for a moment, it’s like having a very fast automatic lane correction assistant which makes corrections after the car hits those small bumps, keeping the vehicle on course.

Through this innovative approach, we leverage the principles of QEC at the individual qubit level. Specifically, we have stored logical information nonlocally in the phase space of a harmonic oscillator, which is subject only to locally correlated noise. This has successfully improved the lifetime of qubits at the individual level, showcasing an approach through which qubit performance can be improved with built-in hardware error correction.

To bridge the gap to full-fledged quantum processing, combining this inner GKP-hardware code with a more conventional outer QEC code will be necessary. On this path, the improvements from the inner code will relax the constraints on the daunting number of qubits required for computation with the outer code. Because far fewer qubits are required to execute Nord Quantique’s hardware to QEC when compared to a processor that relies only on the conventional approach, this approach promises a far shorter path to fault tolerance, both in terms of sheer number of qubits as well as time and cost of development. 

While there is still a long way to go until fault-tolerant processors are readily available, we believe that QEC is the only viable path towards achieving fault tolerance.