As it turns out, there is a quantitative limit to how accurately a code can correct against errors if it is covariant with respect to a continuous symmetry, represented by the following equation: Namely, we consider the situation where the code does not have to correct each error exactly, but only has to reconstruct a good approximation of the logical information. How is that possible, you ask? How do we transition from the microscopic regime, where covariant codes are ruled out for continuous symmetries, to the macroscopic regime, where they are allowed? We provide an answer by resorting to approximate quantum error correction. ![]() Interestingly, however, there’s a loophole: If we consider macroscopic systems, such as a particle with a very large value of spin, then it becomes possible again to construct codes that are covariant with respect to continuous transformations. In other words, the computations we can perform using this scheme are very limited because we can’t perform any continuous symmetry transformation. The advantage of this scheme is that the logical information is never exposed and remains protected all along the computation.īut here’s the catch: Eastin and Knill famously proved that error-correcting codes can be at most covariant with respect to a finite set of transformations, ruling out universal computation with transversal gates. A covariant code allows to perform the transformation directly on the physical qubits, without having to decode the information first: If my information is stored in an encoded form, then in principle I first need to decode the information to uncover the original logical information, apply the flip operation, and then re-encode the new logical information back onto the physical qubits. Suppose I would like to flip my qubit from “0” to “1” and from “1” to “0”. This property ensures that if I apply a symmetry transformation on the logical information, this is equivalent to applying corresponding symmetry transformations on each of the physical systems. Covariant codes for quantum computationĪ code that is compatible with respect to a physical symmetry is called covariant. In this way, instead of storing the actual information we care about on a single qubit, we use extra qubits which we prepare in a complicated state that is designed to protect this information from the noise. Quantum error-correcting codes are particularly promising for quantum computing, since qubits tend to lose their information really fast (current typical ones can hold their information for a few seconds). For instance, if I am holding an atom in my hand (more realistically, it’ll be confined in some fancy trap with lots of lasers), then I can rotate it around and about in space:īy cleverly distributing the information that we care about over several physical systems, an error-correcting code is able to successfully recover the original logical information even if the physical systems are exposed to some noise. What can we say about a quantum error-correcting code that conforms to a physical symmetry? Surprisingly, a continuous symmetry prevents the code from doing its job: A code can conform well to the symmetry, or it can correct against errors accurately, but it cannot do both simultaneously.īy a continuous symmetry, we mean a transformation that is characterized by a set of continuous parameters, such as angles. In a recent contribution, my collaborators and I took a shot at combining the two physical concepts of quantum error correction and physical symmetries. More recently, machine learning tools have been combined with many-body physics to find new ways to identify phases of matter, and ideas from quantum computing were applied to Pozner molecules to obtain new plausible models of how the brain might work. ![]() Put together information theory with quantum mechanics and you’ve opened a whole new field of quantum information theory. Physicists love to draw connections between distinct ideas, interconnecting concepts and theories to uncover new structure in the landscape of scientific knowledge. ![]() It’s always exciting when you can bridge two different physical concepts that seem to have nothing in common-and it’s even more thrilling when the results have as broad a range of possible fields of application as from fault-tolerant quantum computation to quantum gravity.
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