Accurate, nontrivial quantum operations on many qubits are experimentally challenging. As opposed to the standard approach of compiling larger unitaries into sequences of 2-qubit gates, we propose a protocol on Hamiltonian control fields which implements highly selective multi-qubit gates in a strongly-coupled many-body quantum system. We exploit the selectiveness of resonant driving to exchange only 2 out of @math eigenstates of some background Hamiltonian, and discuss a basis transformation, the eigengate, that makes this operation relevant to the computational basis. The latter has a second use as a Hahn echo which undoes the dynamical phases due to the background Hamiltonian. We find that the error of such protocols scales favourably with the gate time as @math , but the protocol becomes inefficient with a growing number of qubits N. The framework is numerically tested in the context of a spin chain model first described by Polychronakos, for which we show that an earlier solution method naturally gives rise to an eigengate. Our techniques could be of independent interest for the theory of driven many-body systems.
The most obvious competitor of our protocol is conventional compiling of any quantum operation into a universal set of single- and two-qubit gates. Extensive research efforts have greatly optmized compiling methods, and in the asymptotics of many qubits, compiling approach becomes increasingly favorable compared to our proposal. For a recent overview, see Ref. @cite_27 . We present our work not as an alternative to compiling, but rather as a creative twist to the fields of condensed matter and quantum control, which might find applications on highly specialized systems. We also present our methods, such as the eigengate presented in Sec. , as tools that may find applications elsewhere.
@cite_27
2801305581 2121957834 1980534149 2079729767
We construct quantum circuits which exactly encode the spectra of correlated electron models up to errors from rotation synthesis. By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity @math where @math is an absolute sum of Hamiltonian coefficients and @math is target precision. For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T gate complexity @math where @math is number of orbitals in the basis. Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime. Compiling to surface code fault-tolerant gates and assuming per gate error rates of one part in a thousand reveals that one can error correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only about one million superconducting qubits in a matter of hours. Standard security proofs of quantum-key-distribution (QKD) protocols often rely on symmetry arguments. In this paper, we prove the security of a three-state protocol that does not possess rotational symmetry. The three-state QKD protocol we consider involves three qubit states, where the first two states @math and @math can contribute to key generation, and the third state @math is for channel estimation. This protocol has been proposed and implemented experimentally in some frequency-based QKD systems where the three states can be prepared easily. Thus, by founding on the security of this three-state protocol, we prove that these QKD schemes are, in fact, unconditionally secure against any attacks allowed by quantum mechanics. The main task in our proof is to upper bound the phase error rate of the qubits given the bit error rates observed. Unconditional security can then be proved not only for the ideal case of a single-photon source and perfect detectors, but also for the realistic case of a phase-randomized weak coherent light source and imperfect threshold detectors. Our result in the phase error rate upper bound is independent of the loss in the channel. Also, we compare the three-state protocol with the Bennett-Brassard 1984 (BB84) protocol. For the single-photon source case, our result proves that the BB84 protocol strictly tolerates a higher quantum bit error rate than the three-state protocol, while for the coherent-source case, the BB84 protocol achieves a higher key generation rate and secure distance than the three-state protocol when a decoy-state method is used. We present a technique for proving the security of quantum-key-distribution (QKD) protocols. It is based on direct information-theoretic arguments and thus also applies if no equivalent entanglement purification scheme can be found. Using this technique, we investigate a general class of QKD protocols with one-way classical post-processing. We show that, in order to analyze the full security of these protocols, it suffices to consider collective attacks. Indeed, we give new lower and upper bounds on the secret-key rate which only involve entropies of two-qubit density operators and which are thus easy to compute. As an illustration of our results, we analyze the Bennett-Brassard 1984, the six-state, and the Bennett 1992 protocols with one-way error correction and privacy amplification. Surprisingly, the performance of these protocols is increased if one of the parties adds noise to the measurement data before the error correction. In particular, this additional noise makes the protocols more robust against noise in the quantum channel. We investigate a general class of quantum key distribution (QKD) protocols using one-way classical communication. We show that full security can be proven by considering only collective attacks. We derive computable lower and upper bounds on the secret-key rate of those QKD protocols involving only entropies of two-qubit density operators. As an illustration of our results, we determine new bounds for the Bennett-Brassard 1984, the 6-state, and the Bennett 1992 protocols. We show that in all these cases the first classical processing that the legitimate partners should apply consists in adding noise.