Classical random walks on finite graphs have an underrated property: a walk from any vertex can reach every other vertex in finite time, provided they are connected. Discrete-time quantum walks on finite connected graphs however, can have infinite hi
tting times. This phenomenon is related to graph symmetry, as previously characterized by the group of direction-preserving graph automorphisms that trivially affect the coin Hilbert space. If a graph is symmetric enough (in a particular sense) then the associated quantum walk unitary will contain eigenvectors that do not overlap a set of target vertices, for any coin flip operator. These eigenvectors span the Infinite Hitting Time (IHT) subspace. Quantum states in the IHT subspace never reach the target vertices, leading to infinite hitting times. However, this is not the whole story: the graph of the 3D cube does not satisfy this symmetry constraint, yet quantum walks on this graph with certain symmetric coins can exhibit infinite hitting times. We study the effect of coin symmetry by analyzing the group of coin-permutation symmetries (CPS): graph automorphisms that act nontrivially on the coin Hilbert space but leave the coin operator invariant. Unitaries using highly symmetric coins with large CPS groups, such as the permutation-invariant Grover coin, are associated with higher probabilities of never arriving, as a result of their larger IHT subspaces.
Quantum walks on lattices can give rise to relativistic wave equations in the long-wavelength limit, but going beyond the single-particle case has proven challenging, especially in more than one spatial dimension. We construct quantum cellular automa
ta for distinguishable particles based on two different quantum walks, and show that by restricting to the antisymmetric and symmetric subspaces, respectively, a multiparticle theory for free fermions and bosons in three spatial dimensions can be produced. This construction evades a no-go theorem that prohibits the usual fermionization constructions in more than one spatial dimension. In the long-wavelength limit, these recover Dirac field theory and Maxwell field theory, i.e., free QED.
We consider the discrimination of two pure quantum states with three allowed outcomes: a correct guess, an incorrect guess, and a non-guess. To find an optimum measurement procedure, we define a tunable cost that penalizes the incorrect guess and non
-guess outcomes. Minimizing this cost over all projective measurements produces a rigorous cost bound that includes the usual Helstrom discrimination bound as a special case. We then show that nonprojective measurements can outperform this modified Helstrom bound for certain choices of cost function. The Ivanovic-Dieks-Peres unambiguous state discrimination protocol is recovered as a special case of this improvement. Notably, while the cost advantage of the latter protocol is destroyed with the introduction of any amount of experimental noise, other choices of cost function have optima for which nonprojective measurements robustly show an appreciable, and thus experimentally measurable, cost advantage. Such an experiment would be an unambiguous demonstration of a benefit from nonprojective measurements.
Large weak values have been used to amplify the sensitivity of a linear response signal for detecting changes in a small parameter, which has also enabled a simple method for precise parameter estimation. However, producing a large weak value require
s a low postselection probability for an ancilla degree of freedom, which limits the utility of the technique. We propose an improvement to this method that uses entanglement to increase the efficiency. We show that by entangling and postselecting $n$ ancillas, the postselection probability can be increased by a factor of $n$ while keeping the weak value fixed (compared to $n$ uncorrelated attempts with one ancilla), which is the optimal scaling with $n$ that is expected from quantum metrology. Furthermore, we show the surprising result that the quantum Fisher information about the detected parameter can be almost entirely preserved in the postselected state, which allows the sensitive estimation to approximately saturate the optimal quantum Cram{e}r-Rao bound. To illustrate this protocol we provide simple quantum circuits that can be implemented using current experimental realizations of three entangled qubits.
We review an approach to fault-tolerant holonomic quantum computation on stabilizer codes. We explain its workings as based on adiabatic dragging of the subsystem containing the logical information around suitable loops along which the information remains protected.
We present a general theory of entanglement-assisted quantum convolutional coding. The codes have a convolutional or memory structure, they assume that the sender and receiver share noiseless entanglement prior to quantum communication, and they are
not restricted to possess the Calderbank-Shor-Steane structure as in previous work. We provide two significant advances for quantum convolutional coding theory. We first show how to expand a given set of quantum convolutional generators. This expansion step acts as a preprocessor for a polynomial symplectic Gram-Schmidt orthogonalization procedure that simplifies the commutation relations of the expanded generators to be the same as those of entangled Bell states (ebits) and ancilla qubits. The above two steps produce a set of generators with equivalent error-correcting properties to those of the original generators. We then demonstrate how to perform online encoding and decoding for a stream of information qubits, halves of ebits, and ancilla qubits. The upshot of our theory is that the quantum code designer can engineer quantum convolutional codes with desirable error-correcting properties without having to worry about the commutation relations of these generators.
We show how to protect a stream of quantum information from decoherence induced by a noisy quantum communication channel. We exploit preshared entanglement and a convolutional coding structure to develop a theory of entanglement-assisted quantum conv
olutional coding. Our construction produces a Calderbank-Shor-Steane (CSS) entanglement-assisted quantum convolutional code from two arbitrary classical binary convolutional codes. The rate and error-correcting properties of the classical convolutional codes directly determine the corresponding properties of the resulting entanglement-assisted quantum convolutional code. We explain how to encode our CSS entanglement-assisted quantum convolutional codes starting from a stream of information qubits, ancilla qubits, and shared entangled bits.
We investigate the construction of quantum low-density parity-check (LDPC) codes from classical quasi-cyclic (QC) LDPC codes with girth greater than or equal to 6. We have shown that the classical codes in the generalized Calderbank-Shor-Steane (CSS)
construction do not need to satisfy the dual-containing property as long as pre-shared entanglement is available to both sender and receiver. We can use this to avoid the many 4-cycles which typically arise in dual-containing LDPC codes. The advantage of such quantum codes comes from the use of efficient decoding algorithms such as sum-product algorithm (SPA). It is well known that in the SPA, cycles of length 4 make successive decoding iterations highly correlated and hence limit the decoding performance. We show the principle of constructing quantum QC-LDPC codes which require only small amounts of initial shared entanglement.
We explain how to combine holonomic quantum computation (HQC) with fault tolerant quantum error correction. This establishes the scalability of HQC, putting it on equal footing with other models of computation, while retaining the inherent robustness the method derives from its geometric nature.
We provide several formulas that determine the optimal number of entangled bits (ebits) that a general entanglement-assisted quantum code requires. Our first theorem gives a formula that applies to an arbitrary entanglement-assisted block code. Corol
laries of this theorem give formulas that apply to a code imported from two classical binary block codes, to a code imported from a classical quaternary block code, and to a continuous-variable entanglement-assisted quantum block code. Finally, we conjecture two formulas that apply to entanglement-assisted quantum convolutional codes.
