Quantum Circuits

Abstract—Quantum computing offers immense speedup in performing tasks such as data encryption and searching. The quantum algorithms can be modeled using classical computing devices, however classical computer simulations cannot deal efficiently with the parallelism ...

In this paper, the problem of constructing an efficient quantum circuit for the implementation of an arbitrary quantum computation is addressed. To this end, a basic block based on the cosine-sine decomposition method is suggested which contains $l$ qubits. In addition, a previously proposed quantum-logic synthesis method based on quantum Shannon decomposition is recursively applied to reach unitary gates over $l$ qubits. Then, the basic block is used and some optimizations are applied to remove redundant gates. It is shown that the exact value of $l$ affects the number of one-qubit and CNOT gates in the proposed method. In comparison to the previous synthesis methods, the value of $l$ is examined consequently to improve either the number of CNOT gates or the total number of gates. The proposed approach is further analyzed by considering the nearest neighbor limitation. According to our evaluation, the number of CNOT gates is increased by at most a factor of $\frac{5}{3}$ if the nea...

A superluminal quantum-vortex model of the electron and the positron is produced from a superluminal double-helix model of the photon during electron-positron pair production. The two oppositely-charged (with Q = ±e sqrt (2/α) = 16.6e) open-helix spin-½ half-photons compose the double-helix photon. These half-photons separate and curl up their separated superluminal single-helical trajectories to form an electrically-charged superluminal closed-helix spin-½ quantum-vortex electron model and a corresponding positron model. The helical radius and the Dirac equation's zitterbewegung angular frequency of the quantum vortex electron and positron models equal the helical radius and zitterbewegung angular frequency of the two spin-½ half-photons, each of energy E = mc^2 , that composed the double-helix photon model of energy E = 2mc^2 from which the electron and positron models were produced. The photon and electron models are also compatible when a photon of energy E > 2mc^2 produces a relativistic electron-positron pair. Implications of the quantum vortex electron model for electron stability are discussed.

Because the qubit probabilities must be preserved at the output of the quantum gate, it is noted that all matrices representing them are unitary, ie U*U=I. An important unitary matrix property is that of full rank. This property implies that quantum gate matrix rows and columns are ...

For a long time, one of my dreams was to describe the nature of uncertainty axiomatically, and it looks like I've finally done it in my co∼eventum mechanics! Now it remains for me to explain to everyone the co∼eventum mechanics in the most approachable way. This is what I'm trying to do in this work. The co∼eventum mechanics is another name for the co∼event theory, i.e., for the theory of experience and chance which I axiomatized in 2016 [1, 2]. In my opinion, this name best reflects the co∼event-based idea of the new dual theory of uncertainty, which combines the probability theory as a theory of chance, with its dual half, the believability theory as a theory of experience. In addition, I like this new name indicates a direct connection between the co∼event theory and quantum mechanics, which is intended for the physical explanation and description of the conict between quantum observers and quantum observations [4]. Since my theory of uncertainty satises the Kolmogorov axioms of probability theory, to explain this co∼eventum mechanics I will use a way analogous to the already tested one, which explains the theory of probability as a theory of chance describing the results of a random experiment. The simplest example of a random experiment in probability theory is the " tossing a coin ". Therefore, I decided to use this the simplest random experiment itself, as well as the two its analogies: the " "flipping a coin " and the " spinning a coin " to explain the co∼eventum mechanics, which describes the results of a combined experienced random experiment. I would like to resort to the usual for the probability theory " coin-based " analogy to explain (and first of all for myself) the logic of the co∼eventum mechanics as a logic of experience and chance. Of course, this analogy one may seem strange if not crazy. But I did not come up with a better way of tying the explanations of the logic of the co∼eventum mechanics to the coin-based explanations that are commonly used in probability theory to explain at least for myself the logic of the chance through a simple visual " coin-based " model that clarifies what occurs as a result of a combined experienced random experiment in which the experience of observer faces the chance of observation. I hope this analogy can be useful not only for me in understanding the co∼eventum mechanics.

In circuit QED, protocols for quantum gates and readout of superconducting qubits often rely on the dispersive regime, reached when the qubit-photon detuning {\Delta} is large compared to their mutual coupling strength. For qubits including the Cooper-pair box and transmon, selection rules dramatically restrict the contributions to dispersive level shifts {\chi}. By contrast, without selection rules many virtual transitions contribute to {\chi} and can produce sizable dispersive shifts even at large detuning. We present theory for a generic multi-level qudit capacitively coupled to one or multiple harmonic modes, and give general expressions for the effective Hamiltonian in second and fourth order perturbation theory. Applying our results to the fluxonium system, we show that the absence of strong selection rules explains the surprisingly large dispersive shifts observed in experiments and also leads to the prediction of a two-photon vacuum Rabi splitting. Quantitative predictions from our theory are in good agreement with experimental data over a wide range of magnetic flux and reveal that fourth-order resonances are important for the phase modulation observed in fluxonium spectroscopy.

CORRECTED 2ND DRAFT
The no-signaling arguments for orthodox quantum theory assume a violation of the general action-reaction principle that Einstein used to get from special relativity to general relativity. Special relativity had action of spacetime geometry on the geodesic motion of particles without any direct reaction of those particles back on the spacetime geometry. Similarly, Bohm and Hiley pointed out that the no-signaling property of orthodox quantum theory’s entanglement depended on the Schrodinger quantum information pilot field having “no sources.” The pilot field changed the classical world lines of particles without any direct reaction of those particles back on their quantum information pilot field. Such direct back-reaction would result in stand-alone entanglement signals. These entanglement signals seem to violate special relativity because they appear to be spacelike faster-than-the speed of light. However, R. I. Sutherland, building on the 1950’s work of O. Costa de Beauregard, I.J. Good and Fred Hoyle leading back to Wheeler and Feynman (1940 – 47) has shown that the violation of Bell’s inequality is best understood as local retrocausality that is completely relativistically invariant. John Bell knew about this, but rejected it because it seemed to him to contradict free will.  Yakir Aharonov and his students as well as John Cramer and Hoyle and Narlikar have developed these ideas further since the 1950s up to the present time. Of particular experimental interest is Aephraim Steinberg’s work on photon Bohmian trajectories in space using Aharonov’s post-selected “destiny” and pre-selected “history” “weak measurement” technique. The bottom line here is that real Bohmian particle trajectories can be observed – the variables are no longer “hidden” via Aharonov’s weak measurements. This is the most parsimonious understanding of the facts minimizing the algorithmic complexity in the space of possible interpretations of orthodox quantum theory.

Rod Sutherland has shown that the lack of direct back reaction from particle to wave enforcing the no-signaling property of entanglement is precisely the old de Broglie guidance constraint that Antony Valentini characterizes as “sub-quantal equilibrium” with “signal locality” as a limiting case of a more general post-quantum theory. Hagen Kleinert  has also shown that this limiting case is a WKB approximation. Violation of the de Broglie guidance constraint leads to entanglement signaling that may be equivalent to the super-computation around closed timelike curves discussed by Seth Lloyd and others.  These results also apply to the Dirac fermion and the boson gauge theories in Bohmian form.

The application of post-quantum theory’s violation of the de-Broglie guidance constraint corresponding to Antony Valentini’s “sub-quantum non-equilibrium” “signal nonlocality” to naturally conscious AI machines will also be considered.

This paper breaks new ground. It is first a commentary on the following paper by Rod Sutherland. However, second, I will also consider the extension of quantum theory to the more general post-quantum theory that is not considered except in passing in his paper below.  Post-quantum theory obeying Einstein’s general action-reaction principle corresponds to Antony Valentini’s “sub-quantum non-equilibrium” with entanglement signal nonlocality. That is, we do not need a classical signal key to locally decode a message at Bob’s receiver sub-system encoded in the distant sender Alice’s part of the entangled whole system. The message would be in non-equilibrium non-random statistics of the reduced density matrix of Bob’s receiver controlled by Alice even if Alice is inside Bob’s Lorentz invariant future light cone.

The recent proposal (Planat and Kibler 2008 arXiv:0807.3650 [quant-ph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this communication, the geometry of a Clifford group G is expressed as a BN-pair, i.e. a pair of subgroups B and N that generate G, is such that intersection H = B∩N is normal in G, the group W = N/H is a Coxeter group and two extra axioms are satisfied by the double cosets acting on B. The BN-pair used in this decomposition relies on the swap and match gates already introduced for classically simulating quantum circuits (Jozsa and Miyake 2008 arXiv:0804.4050 [quant-ph]). The two- and three-qubit cases are related to the configuration with 27 lines on a smooth cubic surface.

We investigate stabilizer codes with carrier qudits of equal dimension $D$, an arbitrary integer greater than 1. We prove that there is a direct relation between the dimension of a qudit stabilizer code and the size of its corresponding stabilizer, and this implies that the code and its stabilizer are dual to each other. We also show that any qudit

Theoretical computer scientists have been debating the role of oracles since the 1970's. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linear-sized circuits, by proving a new

We present elements of quantum circuits translations from the stan- dard network (or circuit) model to the one-way one. We present a general translation scheme, give an account of currently existing tools to apply the scheme, and propose an extension of those tools into a complete translation calculus. We analyze the set of difficul- ties incurred from such work, and

A new algorithm for minimization of quantum cost of quantum circuits has been designed. The quantum cost of different quantum circuits of particular interest (eg. circuits for EPR, quantum teleportation, shor code and different quantum arithmetic operations) are computed by using the proposed algorithm. The quantum costs obtained using the proposed algorithm is compared with the existing results and it is found that the algorithm has produced minimum quantum cost in all cases.

Quantum information science has shown that harnessing quantum mechanical effects can dramatically improve performance for certain tasks in communication, computation and measurement. Already a number of photonic quantum circuits have been realized for quantum metrology, lithography and quantum logic gates. However, these demonstrations have relied on large-scale (bulk) optical elements bolted to large optical tables, thereby making them inherently unscalable and confining them to the research laboratory. This paper reports on the implementation of quantum optic integrated circuits, which not only dramatically reduces the footprint of quantum circuits, but allows unprecedented stability and control of the optical path length; this reveals the possibility for realizing previously unfeasible large scale quantum circuits. Results from femtosecond laser directly-written quantum circuits are also discussed. A large range of chip-scale "photonic building blocks" have been realized using this so-called direct-write technique and they include multimode optical interconnects, directional couplers and interferometers. The directly written quantum circuits exhibited high 2 and 3-photon Hong-Ou-Mandel (HOM) interference visibilities of 95.8% plusmn 0.5 and 84% plusmn 3% respectively.

New in the probability theory and eventology theory, the concept of Kopula (eventological copula) is introduced. The theorem on the characterization of the sets of events by Kopula is proved, which serves as the eventological pre-image of the well-known Sclar's theorem on copulas (1959). The Kopulas of doublets and triplets of events are given, as well as of some N-sets of events. For further development of this topic, see my later works: https://www.academia.edu/34390291/, https://www.academia.edu/34373279/, https://www.academia.edu/34357251/

Keywords: eventology, probability, Kolmogorov event, event, set of events, Kopula (eventological copula), Kopula characterizing a set of events.

If you are interested in my work, and you have the opportunity to provide financial support for the publication of my book "Theory of experience and of chance as a theory of being," I ask you to use my PayPal account: paypal.me/EventologyTheory

In the framework of quantum computational tensor network [D. Gross and J. Eisert, Phys. Rev. Lett. {\bf98}, 220503 (2007)], which is a general framework of measurement-based quantum computation, the resource many-body state is represented in a tensor-network form, and universal quantum computation is performed in a virtual linear space, which is called a correlation space, where tensors live. Since any unitary operation, state preparation, and the projection measurement in the computational basis can be simulated in a correlation space, it is natural to expect that fault-tolerant quantum circuits can also be simulated in a correlation space. However, we point out that not all physical errors on physical qudits appear as linear completely-positive trace-preserving errors in a correlation space. Since the theories of fault-tolerant quantum circuits known so far assume such noises, this means that the simulation of fault-tolerant quantum circuits in a correlation space is not so straig...

"It has become increasingly apparent that a number of perplexing issues associated with the interpretation of quantum mechanics are more easily resolved once the notion of retrocausality is introduced. The aim here is to list and discuss various examples where a clear explanation has become available via this approach. In so doing, the intention is to highlight that this direction of research deserves more attention than it presently receives."
Written by Roderick Sutherland, whose formalism is the basis for the solution of the "hard problem" i.e. physical explanation of consciousness and the nano-electronic technology it makes possible.

We consider a quantum circuit in which shift and rotation operations on qubits are performed by swap gates and controlled swap gates. These operations can be useful for quantum computers performing elementary arithmetic operations such as multiplication and a bit-wise comparison of qubits.