Quantum information is intrinsically fragile and tends to be very sensitive to noise from its environment. Dealing with this noise is crucial to building large-scale quantum computers. 'Quantum Error Correction' is the field of research that strives to solve this problem by finding ways to correct errors that affect quantum information. The method of choice for protecting quantum information relies on error-correcting codes.

Bosonic quantum error correction

One of the leading approaches to information protection stores information and performs error correction in quantum systems characterized by continuous degrees of freedom, colloquially known as harmonic oscillators. Such platforms are ubiquitous and include electromagnetic systems -- optical fibers, microwave cavities, free space, and optical cavities -- as well as vibration-based systems -- nano-acoustic cavities and vibrational modes of atoms and molecules.

Information is encoded in quantum superpositions of different classical states of each oscillator.
We focus on states with the same energy, which form a sphere whose dimension increases as we add more oscillators. For example, for a single harmonic oscillator, all states with energy E can be thought of as points on a circle of radius √E (Fig. 1). 

Cat-codes [1] form an example of a family of codes that use single harmonic oscillator spaces. The different information states in cat codes are equal superpositions of the vertices of polygons (Fig. 2). The larger the polygon, the better the protection. But what does protection mean here?

For harmonic oscillators, we care about two kinds of errors that affect the radius and the angle of a point on the circle, respectively:

1. Photon Loss: This means that the oscillators lose some energy in the form of a photon. In our model, this results in the shrinking of the circle's radius (Fig. 1). If the system loses all of its energy, then the circle contracts to a point, and indeed, there is only one harmonic oscillator state with zero energy.
2. Dephasing: This noise can alter the superpositions between different states, leading to a partial or complete loss of quantum information. Visually, this translates to rotating the position of the points on the circle which might affect their relative separations (Fig. 1).

We would like to protect against as many photon losses and as much dephasing as we possibly can. Cat code states built from n-sided polygons can protect against n or fewer photon losses and up to 2π/4n radian dephasing shifts.

We propose with a general framework for quantum codes, termed quantum spherical codes, that combines multiple harmonic oscillators in nontrivial ways (i.e. do something better than using multiple copies of cat-codes for each oscillator).

Quantum spherical codes

In general, equal-energy states of an n-oscillator system can be visualized as points on the surface of a 2n-dimensional hypersphere. Photon-loss errors reduce the radius of the hypersphere and dephasing errors rotate points around on the surface. Extending the geometry of cat codes that embed polygons in a circle on the plane, our approach relies on embedding polytopes in the higher-dimensional hypersphere. 

Examples of useful polytopes include the  Möbius-Kantor polygon, tesseract (a.k.a. the hypercube),  and other highly symmetric "Platonic" solids in higher dimensions. We use multiple copies of these polytopes to encode quantum information. 
Moreover, we make use of complex polytopes, which are highly symmetrical figures embedded in complex hyperspheres. A complex hypersphere of dimension n is equivalent to a real hypersphere of dimension 2n. Since quantum states are more naturally described by complex numbers, using complex polytopes provides a more natural language for the description of our code states. 

Polytopes are highly symmetrical objects with properties desirable in various applications. One such property is their design strength. A design of strength t replaces averages of polynomials of degree t over the entire sphere by the sum of their values over the points of the design [2]. One key insight of the paper implies that protection against losses is guaranteed by the design strength of the polytope involved in the code construction. Increasing the strength of the design enables us to protect against the loss of a larger number of photons. At the same time, protection against dephasing noise is tied to the minimum separation between pairs of points of the polytopes representing different states. In general, polytopes with higher point-wise distances have lower design strength. This translates to a tradeoff between dephasing and photon loss protection. Thus, there is a need to find optimum polytopes which can offer good protection against both types of noise.

Conclusion

We construct 17 new codes from complex polytopes, including two infinite families of codes, as well as 27 new codes from real polytopes embedded into the complex space, including 8 infinite families of codes. We further demonstrate superior performance of some of these codes compared to cat-code based encodings. We believe that the framework developed in this paper will serve as the beginning of a structured way of studying and constructing multi-oscillator encodings.

References

1. Leghtas, Zaki et al., "Hardware-efficient autonomous quantum memory protection." Physical Review Letters 111.12 (2013): 120501. doi.org/10.1103/PhysRevLett.111.120501; arXiv:1207.0679
2. Delsarte, Philippe, Goethals, Jean-Marie, and Seidel, Johan Jacob, "Spherical codes and designs." Geom. Dedicata 6, 363-388 (1977). https://doi.org/10.1007/BF03187604