We study quasi-exact quantum error correcting codes and quantum computation with them. A quasi-exact code is an approximate code such that it contains a finite number of scaling parameters, the tuning of which can flow it to corresponding exact codes
, serving as its fixed points. The computation with a quasi-exact code cannot realize any logical gate to arbitrary accuracy. To overcome this, the notion of quasi-exact universality is proposed, which makes quasi-exact quantum computation a feasible model especially for executing moderate-size algorithms. We find that the incompatibility between universality and transversality of the set of logical gates does not persist in the quasi-exact scenario. A class of covariant quasi-exact codes is defined which proves to support transversal and quasi-exact universal set of logical gates for $SU(d)$. This work opens the possibility of quantum computation with quasi-exact universality, transversality, and fault tolerance.
In this work we introduce two code families, which we call the heavy hexagon code and heavy square code. Both code families are implemented by assigning physical data and ancilla qubits to both vertices and edges of low degree graphs. Such a layout i
s particularly suitable for superconducting qubit architectures to minimize frequency collisions and crosstalk. In some cases, frequency collisions can be reduced by several orders of magnitude. The heavy hexagon code is a hybrid surface/Bacon-Shor code mapped onto a (heavy) hexagonal lattice whereas the heavy square code is the surface code mapped onto a (heavy) square lattice. In both cases, the lattice includes all the ancilla qubits required for fault-tolerant error-correction. Naively, the limited qubit connectivity might be thought to limit the error-correcting capability of the code to less than its full distance. Therefore, essential to our construction is the use of flag qubits. We modify minimum weight perfect matching decoding to efficiently and scalably incorporate information from measurements of the flag qubits and correct up to the full code distance while respecting the limited connectivity. Simulations show that high threshold values for both codes can be obtained using our decoding protocol. Further, our decoding scheme can be adapted to other topological code families.
Quantum information scrambling has attracted much attention amid the effort to reconcile the conflict between quantum-mechanical unitarity and the thermalizaiton-irreversibility in many-body systems. Here we propose an unconventional mechanism to gen
erate quantum information scrambling through a high-complexity mapping from logical to physical degrees-of-freedom that hides the logical information into non-separable many-body-correlations. Corresponding to this mapping, we develop an algorithm to efficiently sample a Slater-determinant wavefunction and compute all physical observables in dynamics with a polynomial cost in system-size. The system shows information scrambling in the quantum many-body Hilbert space characterized by the spreading of Hamming-distance. At late time, we find emergence of classical diffusion dynamics in this quantum many-body system. We establish that the operator-mapping enabled growth in out-of-time-order-correlator exhibits exponential-scrambling behavior. The quantum information-hiding mapping approach may shed light on the understanding of fundamental connections among computational complexity, information scrambling and quantum thermalization.
Qubits strongly coupled to a photonic crystal give rise to many exotic physical scenarios, beginning with single and multi-excitation qubit-photon dressed bound states comprising induced spatially localized photonic modes, centered around the qubits,
and the qubits themselves. The localization of these states changes with qubit detuning from the band-edge, offering an avenue of in situ control of bound state interaction. Here, we present experimental results from a device with two qubits coupled to a superconducting microwave photonic crystal and realize tunable on-site and inter-bound state interactions. We observe a fourth-order two photon virtual process between bound states indicating strong coupling between the photonic crystal and qubits. Due to their localization-dependent interaction, these states offer the ability to create one-dimensional chains of bound states with tunable and potentially long-range interactions that preserve the qubits spatial organization, a key criterion for realization of certain quantum many-body models. The widely tunable, strong and robust interactions demonstrated with this system are promising benchmarks towards realizing larger, more complex systems of bound states.
