We present a quantum error correcting code with dynamically generated logical qubits. When viewed as a subsystem code, the code has no logical qubits. Nevertheless, our measurement patterns generate logical qubits, allowing the code to act as a fault
-tolerant quantum memory. Our particular code gives a model very similar to the two-dimensional toric code, but each measurement is a two-qubit Pauli measurement.
We consider a model of quantum computation using qubits where it is possible to measure whether a given pair are in a singlet (total spin $0$) or triplet (total spin $1$) state. The physical motivation is that we can do these measurements in a way th
at is protected against revealing other information so long as all terms in the Hamiltonian are $SU(2)$-invariant. We conjecture that this model is equivalent to BQP. Towards this goal, we show: (1) this model is capable of universal quantum computation with polylogarithmic overhead if it is supplemented by single qubit $X$ and $Z$ gates. (2) Without any additional gates, it is at least as powerful as the weak model of permutational quantum computation of Jordan[1, 2]. (3) With postselection, the model is equivalent to PostBQP.
We give a procedure for reverse engineering a closed, simply connected, Riemannian manifold with bounded local geometry from a sparse chain complex over $mathbb{Z}$. Applying this procedure to chain complexes obtained by lifting recently developed qu
antum codes, which correspond to chain complexes over $mathbb{Z}_2$, we construct the first examples of power law $mathbb{Z}_2$ systolic freedom. As a result that may be of independent interest in graph theory, we give an efficient randomized algorithm to construct a weakly fundamental cycle basis for a graph, such that each edge appears only polylogarithmically times in the basis. We use this result to trivialize the fundamental group of the manifold we construct.
Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product
and a brute force decoder for the other code. We apply this construction to the specific case of the product of a surface code with a small code such as a $[[4,2,2]]$ code, which we call an augmented surface code. The distance of the augmented surface code is the product of the distance of the surface code with that of the small code, and the union-find decoder, with slight modifications, can decode errors up to half the distance. We present numerical simulations, showing that while the threshold of these augmented codes is lower than that of the surface code, the low noise performance is improved.
We present a quantum LDPC code family that has distance $Omega(N^{3/5}/operatorname{polylog}(N))$ and $tildeTheta(N^{3/5})$ logical qubits. This is the first quantum LDPC code construction which achieves distance greater than $N^{1/2} operatorname{po
lylog}(N)$. The construction is based on generalizing the homological product of codes to a fiber bundle.
Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the mixed phase, a maximally mixed i
nitial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems -- i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears -- purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.
We construct an exactly solvable commuting projector model for a $4+1$ dimensional ${mathbb Z}_2$ symmetry-protected topological phase (SPT) which is outside the cohomology classification of SPTs. The model is described by a decorated domain wall con
struction, with three-fermion Walker-Wang phases on the domain walls. We describe the anomalous nature of the phase in several ways. One interesting feature is that, in contrast to in-cohomology phases, the effective ${mathbb Z}_2$ symmetry on a $3+1$ dimensional boundary cannot be described by a quantum circuit and instead is a nontrivial quantum cellular automaton (QCA). A related property is that a codimension-two defect (for example, the termination of a ${mathbb Z}_2$ domain wall at a trivial boundary) will carry nontrivial chiral central charge $4$ mod $8$. We also construct a gapped symmetric topologically-ordered boundary state for our model, which constitutes an anomalous symmetry enriched topological phase outside of the classification of arXiv:1602.00187, and define a corresponding anomaly indicator.
We analyze the class of Generalized Double Semion (GDS) models in arbitrary dimensions from the point of view of lattice Hamiltonians. We show that on a $d$-dimensional spatial manifold $M$ the dual of the GDS is equivalent, up to constant depth loca
l quantum circuits, to a group cohomology theory tensored with lower dimensional cohomology models that depend on the manifold $M$. We comment on the space-time topological quantum field theory (TQFT) interpretation of this result. We also investigate the GDS in the presence of time reversal symmetry, showing that it forms a non-trivial symmetry enriched toric code phase in odd spatial dimensions.
We construct a three-dimensional quantum cellular automaton (QCA), an automorphism of the local operator algebra on a lattice of qubits, which disentangles the ground state of the Walker-Wang three fermion model. We show that if this QCA can be reali
zed by a quantum circuit of constant depth, then there exists a two-dimensional commuting projector Hamiltonian which realizes the three fermion topological order which is widely believed not to be possible. We conjecture in accordance with this belief that this QCA is not a quantum circuit of constant depth, and we provide two further pieces of evidence to support the conjecture. We show that this QCA maps every local Pauli operator to a local Pauli operator, but is not a Clifford circuit of constant depth. Further, we show that if the three-dimensional QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional QCA acting on fermionic degrees of freedom which cannot be realized by a quantum circuit of constant depth; i.e., we prove the existence of a nontrivial QCA in either three or two dimensions. The square of our three-dimensional QCA can be realized by a quantum circuit of constant depth, and this suggests the existence of a $mathbb{Z}_2$ invariant of a QCA in higher dimensions, totally distinct from the classification by positive rationals (i.e., by one integer index for each prime) in one dimension. In an appendix, unrelated to the main body of this paper, we give a fermionic generalization of a result of Bravyi and Vyalyi on ground states of 2-local commuting Hamiltonians.
Recently we proposed a family of magic state distillation protocols that obtains asymptotic performance that is conjectured to be optimal. This family depends upon several codes, called inner codes and outer codes. We presented some small examples of
these codes as well as an analysis of codes in the asymptotic limit. Here, we analyze such protocols in an intermediate size regime, using hundreds to thousands of qubits. We use BCH inner codes, combined with various outer codes. We extend our protocols by adding error correction in some cases. We present a variety of protocols in various input error regimes; in many cases these protocols require significantly fewer input magic states to obtain a given output error than previous protocols.
