Quantum logic as superbraids of entangled qubit world lines

Transcription

1 PHYSICAL REVIEW A 81, 038 (010) Quantum logic as superbrais of entangle qubit worl lines Jeffrey Yepez Air Force Research Laboratory, Hanscom Air Force Base, Massachusetts 01731, USA (Receive 7 July 009; revise manuscript receive 9 November 009; publishe 5 February 010) Presente is a topological representation of quantum logic that views entangle qubit spacetime histories (or qubit worl lines) as a generalize brai, referre to as a superbrai. The crossing of worl lines can be quantum-mechanical in nature, most conveniently expresse analytically with laer-operator-base quantum gates. At a crossing, inepenent worl lines can become entangle. Complicate superbrais are systematically reuce by recursively applying quantum skein relations. If the superbrai is close (e.g., representing quantum circuits with close-loop feeback, quantum lattice gas algorithms, loop or vacuum iagrams in quantum fiel theory), then one can ecompose the resulting superlink into an entangle superposition of classical links. Thus, one can compute a superlink invariant, for example, the Jones polynomial for the square root of a classical knot. DOI: /PhysRevA PACS number(s): Lx, 0.10.Kn I. INTRODUCTION In topological quantum computing [1,], a quantum gate operation erives from braiing quasiparticles, for example, two Majoranna zero-energy vortices mae of entangle Cooper-pair states in a p + ip superconuctor where the vortex-vortex phase interaction has a non-abelian SU() gauge group [3,4]. Dynamically braiing such quantum vortices (point efects in a planar cross section of the conensate) inuces phase shifts in the quantum flui s multiconnecte wave function. Local nonlinear interactions between eflects (vortex-vortex straining) is otherwise neglecte; that is, the separation istance δ of the zero-moe vortices is much greater then the vortex core size, which scales as the coherence length ξ δ in quantum fluis. The braiing occurs aiabatically so the quantum flui remains in local equilibrium an the number of eflects (qubits) remains fixe. For implementations, the usual question is how can quantum logic gates, an in turn quantum algorithms, be represente by braiing eflects, quasiparticles with a non-abelian gauge group. This article aresses the relate funamental question about the relationship between quantum entanglement, tangle strans, quantum logic, an quantum-information theory [5 11]. How can a quantum logic gate, an in turn a quantum algorithm, be ecompose into a linear combination (entangle superposition) of classical brai operators? The goal is to comprehen an categorize quantum-information topologically. This is one by first viewing a quantum gate as a brai of two qubit spacetime histories or worl lines. Qubit-qubit interaction associate with a quantum gate is renere as a tree-level scattering iagram, a form of ribbon graph. A quantum algorithm may be represente as a weave of such graphs, a superbrai of qubit worl lines. Finally, one closes a superbrai to form a superlink. In fact, quantum lattice gas algorithms, for example, those employe for the simulation of superfluis themselves [1], are a goo superlink archetype; hence the share nomenclature. With this technology we can calculate superlink invariants. In principle, each quantum circuit has its own unique invariant (associate with Laurent series); for example, two competing quantum circuit implementations of a particular algorithm can be juge equivalent, irrespective of circuit schemes an the placement of gates an wires. If two quantum algorithms, firstan secon-orer accurate, are topologically equivalent, then the simpler one can be use for analytical preictions of their common effective theory while the latter can be use for faster simulations with fewer resources. In short, presente is a quantum generalization of the Temperley-Lieb algebra TL Q an Artin brai group B Q :a superbrai an its closure, a superlink, is forme out of the worl lines of Q qubits (strans) unergoing ynamics generate by quantum gates. Furthermore, the superbrai representation of quantum ynamics works equally well for either bosonic or fermionic quantum simulations. There exists a classical limit where the generalize Temperley-Lieb algebra an the superbrai group, efine later in this article, reuce to the usual Temperley-Lieb algebra an brai group. There also exists a purely quantum-mechanical limit where superbrais reuce to conservative quantum logic operators. Thus, the superbrai is the progenitor of the brai operator an quantum gate operator. This article is organize as follows. A conense review of knot theory sufficient to efine the Jones polynomial an a conense review of qubit laer operators sufficient to efine quantum logic operators an superbrai operators are given in Sec. II. A iagrammatic representation of quantum logic, a hybri between the usual quantum circuit iagrams an Feynman iagrams, is given in Sec. III. A generalization of the Temperley-Lieb algebra (that inclues fermionic particle ynamics) is presente in Sec. IV. The superbrai group relations are given in terms of this generalize algebra in Sec. V, whereby the superbrai operator is cast in three ifferent mathematical forms. The first (exponential form) illustrates how a superbrai is an amalgamation of a classical brai operator an a quantum gate. The secon (knot theory form) illustrates the very close connection to the well-known classical brai operator. The thir (prouct form) illustrates the physics of pairwise quantum entanglement as the braiing of two qubit worl lines through a particular quantum gate Euler angle. Quantum skein relations are given in Sec. VI, an calculations of superlink invariants, for example, for the square root knots introuce here, are given in Sec. VII. Thus, using the superbrai formalism, quantum knots such as the square root of the unknot or the square root of the trefoil knot are well-efine topological objects an each have their own knot invariant. Finally, a brief summary /010/81()/038(10) The American Physical Society

2 JEFFREY YEPEZ PHYSICAL REVIEW A 81, 038 (010) an some final remarks are given in Sec. VIII. To help make the presentation more accessible to a wier auience, a primer on joint quantum logic is presente in the Appenix, which helps explain the origin of the superbrai s generator. II. LINKS, LADDERS, AND LOGIC Classical brai operators (nearest-neighbor permutations), represente in terms of Temperley-Lieb algebra [13], were originally iscovere in six-vertex Potts moels an statistical mechanical treatments of two-imensional lattice systems [14,15]. The quantum algorithm to compute the Jones polynomial [16,17] employs unitary gate operators that are mappe to unitary representations of the brai group, that is, generate by Hermitian representations of the Temperley-Lieb algebra. To prepare for our presentation of superbrais as a topological representation of the quantum logic unerlying quantuminformation ynamics, let us first briefly review some basics of knot theory an some basic quantum gate technology using qubit laer operators. A link comprising Q strans, enote by L, say, is the closure of a brai. The Jones polynomial V L (A) is an invariant of L [18], where A is a complex parameter associate with the link whose physical interpretation will be presente later in this article. V L (A) is a Laurent series in A. The Jones polynomial is efine for a link embee in three space an oriente link. One projects L onto a plane. In the projecte image, in general crossing of strans occurs but is isambiguate by its sign ±1; that is, one assigns overcrossings the sign of +1 an unercrossings 1. The writhe w(l) is sum of the signs of all the crossings, that is, the net sign of a link s planar projection. The Jones polynomial is compute as follows: V L (A) = ( A 3 ) w(l) K L(A), (1) where K L (A) is the Kauffman bracket of the link. K L (A) is etermine from a planar projection of L, for example, using the skein relations below. In the simplest case of an unknotte link (or unknot), the Kauffman bracket is = K (A) = = A A. () The Kauffman bracket of a isjoint union of n unknots has the value n, for example, =. K L (A) for a link with crossings can be compute recursively using a skein relation that equates it to the weighte sum of two links, each with one less crossing: = A + A 1, (3a) = A + A 1, (3b) where A an its inverse are the weighting factors. As an example, let us recursively apply (3) to prove an intuitively obvious link ientity = One reuces the relevant brai as follows: (3a) = A + A 1, (4a) (3b) = A + + A 1, (4b) (3b) = A A, (4c) () = A A, (4) = + ( + A + A, (4e) () =. (4f) A quantum gate represents the qubit-qubit coupling that occurs at the crossing of worl lines of a pair of qubits, say, q α an q γ inasystemofq qubits. Every quantum gate is generate by a Hermitian operator, E αγ say, an whose action on the quantum state may be expresse as...q α...q γ... = e iζe αγ...q α...q γ..., (5) where ζ is a real parameter. The archetypal case consiere here is Eαγ = E αγ ; the generator is iempotent. Suppose the system of qubits is employe to moel the quantum ynamics of fermions or bosons. Is there an analytical form of the generator E αγ that allows one to easily istinguish between the two cases? It is natural to begin by treating fermion statistics. With the logical one state of a qubit 1 = ( 0 1), notice that σ z 1 = 1, so one can count the number of preceing bits that contribute to the overall phase shift ue to fermionic bit exchange involving the γ th qubit with tensor prouct operator, γ 1 σz ψ =( 1) N γ ψ. The phase factor is etermine by the number of bit crossings N γ = γ 1 k=1 n k in the state ψ an where the Boolean number variables are n k [0, 1]. Hence, an annihilation operator is ecompose into a tensor prouct known as the Joran-Wigner transformation [19], γ 1 a γ = σz a 1 Q γ, (6) for integer γ [1,Q] an here the singleton operator is a = 1 (σ x + iσ y ), where σ i for i = x,y,z are the Pauli matrices. See page 17 of Ref. [0] for a typical way of etermining N γ. Equation (6) an its transpose, the creation operator a γ = at γ, satisfy the anticommutation relations {a γ,a β }=δ γβ, {a γ,a β }=0, {a γ,a β }=0. (7) The Hermitian generator of a quantum gate can be analytically expresse in terms of qubit creation an annihilation operators. A novel generator that is manifestly Hermitian is the following: E αγ = 1 [ A n α A n γ Aa α a γ A 1 a γ a α + ( 1)n α n γ ], (8) where = A A is real. The parameter is Boolean, an it allows one to select between Fermi ( = 1) or Bose ( = 0) statistics of the moele quantum particles. The operator generate by E αγ is e z E αγ = 1 Q + (e z 1)E αγ, (9) which is a quantum state interchange for the case when z is pure imaginary. That is, (9) is a conservative quantum logic gate for z = iζ. For the case when z is complex, (9) is proportional to a superbrai operator. The origin of the mathematical form of (8) is given in the Appenix. III. DIAGRAMMATIC QUANTUM LOGIC The state evolution (5) by the quantum logic gate (9) can be unerstoo as scattering between two 038-

3 QUANTUM LOGIC AS SUPERBRAIDS OF ENTANGLED... PHYSICAL REVIEW A 81, 038 (010) qubits = A e iζ A + A (eiζ 1), (1c) ψ =e iζ E αγ ψ, (10a) = [1 + (e iζ 1) ] (1) ψ =e iζ E αγ ψ, (10b) where the gauge fiel that couples the external qubit worl lines is represente by an internal ouble wavy line (or ribbon). The external lines either overcross or unercross an are assigne +1 an 1 multiplying the action, that is, ±ζ E.This sign isambiguates between a quantum gate an its ajoint, respectively, as shown in (10a) an (10b). Let us enote a qubit graphically q α u α + α, with complex amplitues constraine by conservation of probability u α + α = 1. Starting, for example, with a separable input state ψ = q α q γ, a scattering iagram is a quantum superposition of four oriente graphs: = u α u γ + u α γ + α u γ + α γ (11) Each oriente scattering graph can be reuce to a quantum superposition of classical graphs, or just a single classical graph, as the case may be. There are four quantum skein relations representing ynamics generate by (8): =, (1a) = A A e iζ + A 1 (e iζ 1), (1b) These are the quantum analog of (3). Ajoint quantum skein relations are obtaine simply by taking ζ ζ in the amplitues in the iagrams in (1). All superbrais can be reuce to a quantum superposition of classical brais. The closure of a superbrai forms a superlink. Hence, a superlink can be reuce to a quantum superposition of classical links an, consequently, for each superlink one can compute an associate invariant, for example, a superposition of Jones polynomials. An example calculation of such invariants is presente in Sec. VII. In the context of quantum-information ynamics, a physical interpretation of the parameter A can be renere as follows. If the strans in L are consiere close spacetime histories of Q qubits (e.g., qubit states evolving in a quantum circuit with close-loop feeback), then the left-han sie of (1) represents a trajectory configuration within a piece of the superlink where entanglement is generate by a qubit-qubit coupling that occurs at a quantum gate (i.e., generalize crossing point). For the one-boy cases (1b) an (1c), the right-han sie represents classical alternatives in quantum superposition: 1 e i ζ ( A e i ζ A e ±i ζ ) is the amplitue for no interaction (nonswapping of qubit states), whereas the amplitue of a SWAP interaction (interchanging of qubit states) goes as 1 A 1 (e iζ 1). As a first example of reucing a superbrai, let us recursively apply (1) to prove an obvious evolution ientity: the composition of a quantum gate with its ajoint is the ientity operator, that is, UU = 1. For simplicity, we start with q α = an q γ =, so the initially oriente superbrai is reuce to a superposition of classical brais as follows: (1b) = A A e iζ + A 1 (e iζ 1) (1b) = A A e iζ A A e iζ + A A e iζ A 1 (e iζ 1) + A 1 (e iζ 1) (1c) = A4 + A 4 + cos ζ + (e iζ 1)(e iζ 1) (4) = A4 + A 4 + cos ζ = A 1 (A + A e iζ )(e iζ 1) A 1 (A e iζ A e iζ A + A ) A 1 (A e iζ + A )(e iζ 1) + cos ζ (13) 038-3

4 JEFFREY YEPEZ PHYSICAL REVIEW A 81, 038 (010) It is easy to verify that the same result occurs for inputs q α = an q γ =. Furthermore, the ientity trivially follows for q α = an q γ = an for q α = an q γ = since is Boolean. IV. GENERALIZATION OF TL Q () With ajacent inices, for example, γ = α + 1in(9), we nee write the first inex only (i.e., suppress the secon inice), E α E α,α+1. Using this compresse notation, (8) satisfies the following generalize Temperley-Lieb algebra: E α = E α, α = 1,,...,Q 1, (14a) E α E α±1 E α E α±1 E α E α±1 = E α E α±1, (14b) E α E β = E β E α, α β. (14c) To help unerstan this algebra, we may write (14b) asfollows: E α E α+1 E α E α = X α,α+1, (15a) E α+1 E α E α+1 E α+1 = Y α,α+1, (15b) where X α,α+1 an Y α,α+1 are introuce solely for the purpose of separating (14b) into two equations. For (15) to be equivalent to (14b), one must emonstrate that X α,α+1 = Y α,α+1. Inserting (8) into the left-han sie of (15), after consierable laer operator algebra, one fins that the ifference of the right-han sie of (15) is X α,α+1 Y α,α+1 = ( 1)[(A 4 A 4 )n α n α+1 n α+ A 4 n α n α+1 + A 4 n α+1 n α+ ], (16) vanishing for Boolean. Thus,(14b) follows from (8). One fins X an Y are proportional to, so a remarkable reuction of (14) occurs for the = 0 case: E0α = E 0α, α = 1,,...,Q 1, (17a) (15) E 0α E 0α±1 E 0α = E 0α, (17b) E 0α E 0β = E 0β E 0α, α β. (17c) This is the Temperley-Lieb algebra over a system of Q qubits (TL Q ). Thus, entangle bosonic states generate by E 0α are isomorphic to links generate by E 0α.So(14) isa generalization of TL Q. We now consier the generalize brai that it generates: a superbrai. V. SUPERBRAID GROUP A general superbrai operator is an amalgamation of both a classical brai operator an a quantum gate, b αβ s Aez E αβ, (18) where A an z are complex parameters. Equation (18) can be applie to any two qubits, α an β, in a system of qubits (i.e., we o not impose a restriction to the ajacency case when β = α + 1). Equation (18) can be written in several ifferent ways, each way useful in its own right. Letting z iζ + ln τ, the superbrai operator has the following exponential form b s αβ τ 1 4 e (iζ+ln τ) E αβ = τ 1 4 (e iζ τ) E αβ, (19) where A τ 1 4 (an so τ = A 4 ). The superbrai operator can be written linearly in its generator, b αβ s = A[1 Q + (A 4 e iζ 1)E αβ ]. (0) Thus, the superbrai operator an its inverse can be expresse in knot theory form, ) E αβ, (1a) b s αβ = A 1 Q + A 1 ( 1 e iζ τ 1 + τ ( b s αβ ) 1 = A 1 1 Q + A ( e iζ + τ 1 + τ ) E αβ. (1b) A nontrivial classical limit of quantum logic gates represente as (9) occurs at ζ = π (SWAP operator). Consequently, the superbrai operator in prouct form is b αβ s τ 1 4 e (ln τ+iπ) E αβ e (iζ iπ) E αβ, (a) = b αβ e i(ζ π) E αβ, (b) where b αβ = τ 1 4 e (ln τ+iπ) E αβ is the conventional brai operator. Equation (b) is useful for comprehening the physical behavior of the superbrai operator. It classically brais worl lines α an β an quantum-mechanically entangles these worl lines accoring to the eficit angle ζ π. The superbrai group is efine by bα s bs β = bs β bs α for α β > 1, (3a) bα s bs α+1 bs α + γbs α = bs α+1 bs α bs α+1 + γbs α+1 for 1 α<q, (3b) where γ is a constant that epens on the representation. For (8), we have γ = (A 4 + A 4 e iζ )(1 + e iζ )A. In the classical limit ζ = π, the superbrai operator reuces to the classical brai operator, b α bα s (π, τ), an (3) reuces to the Artin brai group, b α b β = b β b α for α β > 1, (4a) b α b α+1 b α = b α+1 b α b α+1 for 1 α<q. (4b) Equation (4) follows from Eq. (3) because γ = 0 for ζ = π. Also, in this classical limit, Eq. (0) reuces to the brai operator b α = A 1 Q + A 1 E αβ, (5) for α = 1,,...,Q 1 (an technically the stanar brai operator when β = α + 1). After some laer operator manipulations using (8), one fins that (5) b α b α+1 b α b α+1 b α b α+1 = A 1 (A 4 A 4 ) ( 1) (1 n α )n α+1 n α+, (6) where n α a α a α. Since is Boolean, the right-han sie vanishes, an this is just (4b). VI. QUANTUM SKEIN RELATIONS The skein relations (3) for irecte strans are = A = A 1, (7a) = A 1 = A (7b) 038-4

5 QUANTUM LOGIC AS SUPERBRAIDS OF ENTANGLED... PHYSICAL REVIEW A 81, 038 (010) The writhe of a brai, +1 for(7a) an 1 for(7b), is etermine by applying the right-han rule at the crossing point (viz., consiering the strans as vectors), rotating the stran above towar the one below with the axis of rotation either out of the plane (+1) or into the plane ( 1). By inserting (7) into(1), one arrives at a remarkably simple form of the quantum skein relations: =, (8a) (7b) = + (eiζ 1) (7a) = + (eiζ 1), (8b), (8c) = + (e iζ 1) (8) Equation (8) is most useful for reucing a close quantum circuit into a superposition of oriente links. The relations in the one-boy sector can be written as [ ] (8b) A = A + A 1 A (ez 1), (9a) [ ] A 1 (8c) = A 1 + A A (e z 1) (9b) for the case of a complex rotation angle (i.e., iζ z) an where we have multiplie through by A an A 1, respectively, an taking the ajoint of the latter relation. Now with e z = A 4 e iζ an A 4 = τ, wearriveat ( 1 e A = A + A 1 iζ ) τ, (30a) 1 + τ ( e A 1 = A 1 iζ ) + τ + A (30b) 1 + τ the iagrammatic representation of the superbrai operators (1). Thus, we have the corresponence b s αβ A (31) which is just (18) in graphical form. The superbrai operator (31) is unitary when A = 4 1, in which case it reuces to a conservative quantum logic gate. 1 VII. SUPERLINK INVARIANTS Let us begin by writing the conservative quantum logic gate (9) graphically: = + (ez 1) (3) 1 Since flipping over a brai preserves the writhe of its crossing, we may flip over each iagram in (30) to have yet another way to specify operative quantum skein relations. Using this specification, if we take ζ = π, then the flippe-over quantum skein relations just reuce to the classical skein relations (7). where we choose to use the complex time parameter z. Now w number of successive superbrais is = + (ewz 1) (33) where the right-han sie follows from (3) by taking z wz. Letting b 1 K L (A), where L is the closure of b, the Markov trace closure of (33), here enote with angle brackets, is = e wz (34a) = 1 ( 1 ) + e wz = e wz. (34b) Multiplying through by A w, we then have the trace closure of w superbrais: (b s ) w (31) = A w ( 1 ) + 1 (Ae z ) w. (35) In the classical limit ζ = π an e z = A 4, then (35) becomes a stanar w brai: b w (31) = A w ( 1 ) + 1 ( A 3 ) w (36a) = ( A 3 ) w 1 [1 + ( A 4 ) w ( 1)]. (36b) The Jones polynomial is V L (A) (1) = ( A 3 ) w b w,so V L (A) = (A 6 ) w 1 [1 + ( A 4 ) w ( 1)]. (37) For example, consiering classical links forme from the closure of two strans braie an integer number of times with w = 0, 1,, 3, 4, 5...,(37) gives V (A) = A A, (38a) V (A) = A 3, (38b) V (A) = A 4 A 4, (38c) V (A) = A 7 A 3 A 5, (38) V (A) = A 10 + A 6 A A 6, (38e) V (A) = A 13 A 9 + A 5 A 1 A 7, (38f). Equation (38b) is the Jones polynomial invariant for the unknot, (38c) for the Hopf link, (38) for the trefoil knot, an so forth, an these Laurent series are well known. Yet the formula (37) follows from the quantum logic gate relation (34), where w is a time scaling factor. Since time is a continuous variable in quantum logic, we are free to take w to be a real-value parameter whereby the formula for the invariant Jones polynomial remains physically well efine. Thus, for example, we can evaluate (37) for half-integer w, an calculate Jones polynomial invariants for quantum links that are halfway between classical links, which is to say in 038-5

6 JEFFREY YEPEZ PHYSICAL REVIEW A 81, 038 (010) equal superposition of two classical links. For quantum links forme from the closure of two strans braie an half-integer number of times with w = 1, 3, 5, 7...,(37) then gives V (A) = i A A A 6 (39a) A 3 (A + A ) V (A) = ia 3 A 1 A 3 A 7 A 3 (A + A ) V (A) = 1 ia 6 A 4 A 8 A 3 (A + A ) V (A) = ia 9 A A 5 A 9 A 3 (A + A ) (39b) (39c) (39). Equation (39a) is the superlink invariant for the square root of the unknot ( ), (39b) for the square root of the trefoil knot ( ), an so forth. The square root of unknot an trefoil knots are examples of quantum knots, a special class of superlinks. VIII. CONCLUSION Einstein, Poolsky, an Rosen iscovere nonlocal quantum entanglement over three quarters of a century ago [1]. Although this inscrutable seminal result is the most cite one in the physics literature, quantum entanglement still remains one of the most mysterious properties of quantum physics. Here we have strive to unravel some of the mystery behin this important physical effect by renering quantum entanglement geometrically as tangles of the most basic of strans: quantum-informational spacetime trajectories (or qubit worl lines). The avantage of this approach is that it allows us to represent quantum entangle states in terms of intuitive constructs borrowe from the mathematics evelope to unerstan knots. In knot theory, the most funamental construct is braiing (or crossing) two ajacent strans. In quantum-information theory, a funamental construct is entangling two qubits. In general, the braiing operation is nonunitary, whereas an entangling operation (two-qubit universal quantum gate) is manifestly unitary. Yet, these two operations are not entirely unrelate they are in fact special aspects of a general operation, terme a superbrai. The superbrai covers both nonunitary an unitary funamental physical operations. It both brais qubit worl lines an entangles the qubits an in this way mathematically isambiguates these most basic physical processes. Quantum computing algorithms can be specifie in terms of entangling quantum gates that act only between ajacent qubits two-qubit entangling gate operations between nonajacent qubits (customarily use in specifying quantum algorithms in the quantum computing literature) can each be represente as a sequence of two-qubit gate operations acting on ajacent qubit pairs. Thus, local brai an quantum gate operations are both universal operations in their respective contexts. Analytical efining relations for a superbrai operator were presente, as was the algebra for its generator. For Q number of qubits, the generator of a superbrai was foun to be a Hermitian operator that is a generalization of the usual generators in knot theory satisfying the Temperley-Lieb algebra, TL Q (). The generalization presente here hanles both the fermionic an the bosonic cases of quantum-information ynamics; the quantum particles (whose motion efines the quantum-informational strans) can obey either Fermi or Bose statistics. In the bosonic case, the operative generators are the usual ones that satisfy TL Q () an they serve as a Hermitian representation. The generalization of TL Q () was actually neee to hanle the case of entangle fermionic worl lines. These generators, an in turn their respective superbrai operators, are analytically expresse in terms of the qubit laer operators: qubit anticommuting creation an estruction operators an number operators. With the technology presente, one can topologically classify close-loop quantum circuits with various schemes for quantum wires crosse (braie) at some locations an couple together at some locations via quantum logic gates, for example, a sequence of brai operations (particle motion) an quantum gate operations (particle-particle interactions) that specify the local ynamics of a quantum lattice gas use for a computational physics application. The closure of a sequence of superbrais is calle a superlink. We have emonstrate how a superlink invariant may be compute, for example, as a generalize Jones polynomial invariant. Invariants were calculate for two-strane superlinks, an extening this to Q strans is straightforwar. The approach is to reuce a superbrai with n crossings to a simpler superbrai with n 1 crossings by applying a quantum skein relation, a straightforwar generalization of the skein relations of knot theory. The quantum skein relations are summarize as follows. The A parameter commonly use in the classical skein relation of knot theory, = A + A 1 a version of (7a) that is flippe over, may be unerstoo in the context of quantum-information processing as representing the two alternatives for the exchange of a pair of bits for configurations with one bit up (logical zero) an the other bit own (logical one). The iagrammatic convention has information flowing from top to bottom, entering in the top leas of the iagram an exiting from the bottom leas. So if the initial state is......, then the final-state alternatives are: (a) no interaction (ientity operation) or (b) an exchange interaction (classical SWAP) Ais the amplitue for the ientity transition , whereas A 1 represents the amplitue for the exchange transition Since the brai operator conserves bit number an A is an amplitue, we generalize the classical skein relation by allowing the interaction alternative (b) to represent a conservative quantum-mechanical exchange one efine by a bit-conserving, two-qubit entangling gate operation. This resulte in the following quantum skein relation for a superbrai operator: A = A + A 1 [(e z 1)/(A )] 038-6

7 QUANTUM LOGIC AS SUPERBRAIDS OF ENTANGLED... PHYSICAL REVIEW A 81, 038 (010) where = A A. Thus, a classical point occurs for any value of z that causes the quantity in the square bracket to become unity, e z 1 = A. That is, a superbrai reuces toabraiwhene z = A 4. In the case when e z = e iζ A 4 an A is complex unimoular, the phase of A physically acts as an internal ebit phase angle; that is, A = e i(ξ π ) as is iscusse in the Appenix. In (b) we wrote the superbrai operator as the prouct of a brai an conservative quantum gate. It brais two-qubit worl lines an entangles them accoring to the eficit angle ζ = ζ ζ 0, where ζ is the real-value time (relate to the imaginary part of z) parametrizing the operation an ζ = π is the classical point. Entangle qubit worl lines an tangle strans are relate through their respective skein relations sharing a common A parameter. The approach of representing quantum-information topologically in terms of tangle strans [5 11,], an that we have explore here, offers insights about quantum entanglement as the quantum skein relation just mentione, for purely imaginary z, isan entangling conservative quantum gate. The aforementione consierations naturally lea to some relevant further outlooks, for example the observability of quantum knots such as the square root of a knot, which of course has no classical counterpart. Just how the square root of a knot may be physically realize in an experimental setup is not presently known for certain, but accoring to Gell-Mann s totalitarian principle it shoul be experimentally compulsory as a physical phenomenon. Here is one possibility: a superbrai coul exist within a Bose-Einstein conensate (BEC) superflui as a superposition of quantum vortex loops. Furthermore, such vortex loops coul represent a topologically protecte qubit, an entangle states of such qubits coul exist within a spinor BEC. 3 In a BEC, all the vorticity in the flow is pinne to filamentary topological efects in the phase of the conensate, that is, quantum vortices with integer wining numbers [5]. In a spinor BEC with two or more components in the conensate, each component may have its own quantum vortices. In the ilute vortex limit for each component, these quantum vortices act like strans that may be braie an entangle as the spinor superflui flow evolves in time. Consier a configuration of quantum vortices in a superflui comprising two unlinke close loops. 4 Since nearby quantum vortex segments spontaneously unergo reconnection, one woul expect that an so on. To form a qubit, one coul ientify the logical states with the two basic quantum vortex configurations: 0 an 1 So a topologically protecte qubit (superbraie qubit) coul be represente as a superposition of quantum vortex solitons, q =a + b (40) with amplitues constraine by a + b = 1. 5 These amplitues are time-epenent quantities, like the amplitues of a half-integer spin precessing about a uniform magnetic fiel, but in this case the effective Rabi frequency is set by the inverse of two reconnection times. In a two-component BEC, a topologically protecte (perpenicular) pairwise entangle state coul be forme by coupling two superbraie qubits: ψ =ψ 01 +ψ 10. (41) Allowing for spatial overlap of the quantum vortices in each component, the two-qubit quantum vortex configuration coul have its two-loop configuration simultaneously occupy the same location as its unknot quantum vortex configuration in the spinor superflui s secon component. So, might physically occur as (overlappe). If the quantum vortex solitons can be spatially correlate in this way, then the quantum particles comprising the conensate can become physically entangle across their respective vortex cores, an in turn so too the vortex solitons themselves a physical pathway whereby linkage may be relate to entanglement. 6 Superbrai solitons such as (40) coul store topologically protecte qubits, an superlinks such as (41) coul process topologically protecte e-bits. If many superbraie qubits were couple together to realize controllable quantum logic operations, then topological quantum computation may be irectly achievable within spinor superfluis. This offers us a potential alternative to exotic non-abelian vortices (Fibonacci anyons) recently propose for thin-film superconuctor-base topological quantum-information processing. This proposition will be explore in future work on topological quantum computing with superbrais. Finally, since a superbrai can be a nonunitary operation for certain values of the complex time parameter z, in a future stuy one coul also explore if, an if so how, this particular feature might be akin to other behaviors of quantum systems in the real worl. Quantum ynamics is effectively nonunitary because of ecoherence (loss of phase coherency between previously correlate qubits) as well as the nonunitarity of projective measurement (collapse of qubit states onto a logical basis). So, for example, the behavior of the nonunitary ynamics of superbrais coul also be explore regaring its potential connection to measurementbase quantum computation either ruling out or eluciating this connection. 3 Spinor BECs have been realize in a confine col spinor atomic BEC with several hyperfine states [3,4]. 4 The vorticity points along a quantum vortex line, so these act like irecte strans. For simplicity, we omit arrow labels. Furthermore, these loops are not necessarily coplanar, for example, with the first loop in the plane of the paper, the secon one coul be rotate about the axis along its horizontal iameter, an perpenicularly oriente after a 90 rotation. 5 The square root of unknot is realize in this qubit when a = b = 1/, or 45 polarization in the logical basis. 6 This is allowe so long as the separation δ between the component vortices is much less than the coherence length ξ. This particular conition δ ξ for interspinor superflow with quantum vortex solitons of unit wining number is ifferent than the requirement ξ δ in topological quantum computing with quantum logic represente in terms of braie zero-moe vortices in a p + ip superconuctor (as iscusse in the Introuction)

8 JEFFREY YEPEZ PHYSICAL REVIEW A 81, 038 (010) APPENDIX: JOINT QUANTUM LOGIC The coefficients in (8) can be parametrize by a real angle ϑ: E αγ = E αγ (ϑ) with A = cos ϑ + 1 sin ϑ, (A1) an = A A = cscϑ. Then, (8) takes the form αβ : αβ : E αγ = cos ϑ n α + sin ϑ n γ 1 sin ϑ (Aa α a γ αth qubit βth qubit + A 1 a γ a α) + ( 1)n α n γ. (A) The purpose of this appenix is to explain the origin of (A). Let us consier entangling two quantum bits, say, q α an q β, in a system comprise of Q qubits an where the integers α an β [1,Q] are not equal, α β. Joint pair creation an annihilation operators [6] act on a qubit pair, a αβ 1 (a α e iξ a β ), a αβ 1 (a α e iξ a β ), (A3) an are efine in terms of the fermionic laer operators a α an a α. The joint number operator corresponing to (A3)is n αβ a αβ a αβ (A4a) = 1 (n α + n β e iξ a α a β e iξ a β a α), (A4b) where the usual qubit number operator is n α a α a α. Finally, I introuce an entanglement number operator (Hamiltonian) with a simple iempotent form ɛ αβ n αβ + ( 1)n α n β, (A5) where is a Boolean variable (0 for the bosonic case an 1 for the fermionic case). Eq. (A5) acts on some state...q α...q β..., with a qubit of interest locate at α an another at β. For convenience, I will use a shorthan for writing a state, specifying only two qubit locations as subscripts qq αβ...q α...q β..., since the operators act on a qubit pair, regarless of the respective pair s location within the system of Q qubits. 1 Then, the entangle singlet substate is ( ) αβ an the entangle triplet substates are 1 ( ) αβ an 1 ( 11 ± 00 ) αβ. I will refer to the ket qq αβ as perpenicular with respect to its constituent qubits q α an q β when q q an as parallel when q = q. Examples of perpenicular an parallel two-qubit (Fock) states are epicte in Fig. 1. Neglecting normalization factors, the action of (A5)onthe entangle αβ substates is ɛ ( 01 ± 10 ) αβ = 1 (1 e iξ ) 01 1 (1 ± eiξ ) 10, (A6) ɛ ( 11 ± 00 ) αβ = 11. As a matter of convention, the αβ inices are move from ɛ to the state ket upon which the entanglement number operator acts an also the αβ inices are not repeate on the right-han sie of an equation if avoiing reunancy oes not introuce any ambiguity. FIG. 1. Example of perpenicular an parallel two-qubit substates. The perpenicular substate 10 (top pair) an the parallel substate 00 (bottom pair) are epicte with qubits as unit vectors on the complex circle. Two special cases of interest are ξ = π an ξ = 0. For convenience, let us enote these particular angles with plus an minus symbols (+ =π an =0): n ± αβ 1 (a α ± a β )(a α ± a β ), ɛ ± αβ = n± αβ + ( 1)n αn β. (A4 ) Eq. (A4 ) measures entanglement between qubits that are perpenicularly oriente in the four-imensional αβ subspace of the Q -imensional Hilbert space. The sum of these perpenicular joint number operators is relate to the qubit number operators as follows: ɛ + αβ + ɛ αβ = n α + n β + ( 1)n α n β. (A7) The left-han sie of (A7) represents the total information in the αβ subspace spanne by two qubits in the form of quantummechanical e-bits, whereas the right-han sie represents the equivalent amount of information in classical bits. Suppressing inices, ɛ has unity eigenvalue for the singlet state, ɛ ( ) αβ = 01 10, ɛ ( ) αβ = 0, ɛ ( 11 ± 00 ) αβ = 11, (A8a) (A8b) but it has eigenvalue 0 for the triplet state, Conversely, ɛ + has a zero eigenvalue for the singlet state, ɛ + ( ) αβ = 0, ɛ + ( ) αβ = , ɛ + ( 11 ± 00 ) αβ = 11, (A9a) (A9b) but it has eigenvalue 1 for the first triplet state. To avoi any contribution arising from parallel entanglement from the triplet states in (A8b) an (A9b), we must take = 0 in the Hermitian operator (A4 ) to ensure we count only one bit of perpenicular pairwise entanglement. Hence, enoting the pairwise entangle states with qubits of interest at locations α an β as ψ ± ± , then these entangle states are eigenvectors of the = 0 joint number operators (with unit eigenvalue): ɛ ± 0 ψ ± = ψ ±. The parallel joint operators can be efine in terms of the perpenicular joint operators as ε ± 0 1 (1 ɛ+ 0 ± ɛ 0 ±{σ xσ x,ɛ 1 }), (A10) 038-8

9 QUANTUM LOGIC AS SUPERBRAIDS OF ENTANGLED... PHYSICAL REVIEW A 81, 038 (010) suppressing qubit inices, an σ x σ x σ x (α) σ x (β) is shorthan for a tensor prouct. The sum ɛ ɛ 0 + ε+ 0 + ε 0 = 1 (A11) is as funamental to two-qubit substates as the singleton number operator ientity n + n = 1 is to one-qubit states. If the other pairwise entangle states with qubits at α an β are ψ ± ± , then these entangle states are eigenvectors of the joint number operators (with unit eigenvalue): ε ± 0 ψ ± = ψ ±. In summary, the operators ɛ ± 0 an ε± 0 are number operators for the Bell states. A consistent framework for ealing with quantum logic gates using the quantum circuit moel of quantum computation was introuce over a ozen years ago by DiVincenzo et al. [7]. Here we consier an alternative: an analytical approach to quantum computation base on generalize secon quantize operators, which is also practical for numerical implementations. Let us now consier quantum gates that inuce entangle states from previously inepenent qubits. The basic approach employs a conservative quantum logic gate generate by (A5): e iϑ ɛ = 1 + (e iϑ 1)ɛ. (A1) Thus, a similarity transformation of the number operators n α an n β yiels generalize joint number operators n α (ϑ, ξ) eiϑ ɛ αβ n α e iϑ ɛ αβ (A13a) ( ) ( ) ϑ ϑ = cos n α + sin n β + i sin ϑ (e iξ a α a β e iξ a β a α) (A13b) an n β (ϑ, ξ) eiϑ ɛ αβ n β e iϑ ɛ αβ (A14a) ( ) ( ) ϑ ϑ = sin n α + cos n β i sin ϑ (e iξ a α a β e iξ a β a α). (A14b) Thus, the generalize joint number operators rotate continuously from n α (0,ξ) = n α an n β (0,ξ) = n β as ϑ ranges from 0 to π to the number operators n α (π, ξ) = n β an n β (π, ξ) = n α. That information is conserve by this similarity transformation is reaily expresse by the number conservation ientity obtaine by aing (A13a) an (A14): n α (ϑ, ξ) + n β (ϑ, ξ) = n α + n β. (A15) The left-han sie counts the information in its quantummechanical (entangle) form, whereas the right-han sie counts information in its classical form (separable) form. In any case, the total information content in the αβ subspace is conserve. This is why the quantum logic gate operation (A1) is referre to as a conservative operation. Comparing the generalize joint number operator (A13a)to the joint number operator (A4a), we obtain the useful ientity in the special case of maximal entanglement, ( π n α,ξ + π ) ( π = n β,ξ π ) = 1 (n α + n β e iξ a α a β e iξ a β a α) = n αβ. (A16) The tensor prouct n α n β is invariant uner similarity transformation: n α n β = e iϑ ɛ αβ n α n β e iϑ ɛ αβ. (A17) Let us also construct generalize joint laer operators: a α = eiϑ ɛ αβ c α e iϑ ɛ αβ, (A18a) a α = eiϑ ɛ αβ c α e iϑ ɛ αβ. (A18b) After some algebraic manipulation, these can be expresse explicitly just in terms of the original laer operators, [ ( ) ( ) ] ϑ ϑ a α = e i ϑ cos a α + ie iξ sin a β, (A19a) [ ( ) ( ) ] ϑ ϑ a α = ei θ cos a α ie iξ sin a β. (A19b) The prouct of the generalize joint laer operators (A19), n α = a α a α, (A0) yiels the generalize joint number operator (A13a), as expecte. An important special case of (A1) for half angles ξ = π an ϑ = π is an antisymmetric square root of swap gate, U e i π ɛ,π/. (A1) Using (A1) as a similarity transformation, we fin the ientities ɛ + αβ = U n β U an ɛ αβ = U n α U, (A) illustrating how the entanglement operators are relate to the stanar number operators. Noting that the joint number operator is relate to the generalize joint number operator accoring to (A16), an also noting that n α n β is invariant uner the similarity transformation (A17), we can also write a generalization of (A5) as follows: E αβ e iϑ ɛ αβ [n α + ( 1)n α n β ]e iϑ ɛ αβ (A3a) ( ) ( ) (A13b) ϑ ϑ = cos n α + sin n β + i sin ϑ (e iξ a α a β e iξ a β a α) + ( 1)n α n β. (A3b) We have the freeom to choose A = ie iξ, an thus we have recovere the mathematical form of (A). [1] P. Zanari an S. Lloy, Phys. Rev. Lett. 90, (003). [] C. Nayak et al., Rev. Mo. Phys. 80, 1083 (008). [3] D. A. Ivanov, Phys. Rev. Lett. 86, 68 (001). [4] S. Tewari, S. DasSarma, C. Nayak, C. Zhang, an P. Zoller, Phys. Rev. Lett. 98, (007)

10 JEFFREY YEPEZ PHYSICAL REVIEW A 81, 038 (010) [5] L. Kauffman et al.,newj.phys.6, 134 (004). [6] M. Asoueh, V. Karimipour, L. Memarzaeh, an A. T. Rezakhani, Phys. Lett. A37, 380 (004). [7] L. H. Kauffman an S. J. L. Jr., New J. Phys. 4, 73 (00). [8] H. Heyari, J. Phys. A 40, 9877 (007). [9] L. H. Kauffman, Rep. Prog. Phys. 68, 89 (005). [10] H. Dye, Quantum Inf. Process., 117 (003). [11] Y. Zhang, L. Kauffman, an M. Ge, Quantum Inf. Process. 4, 159 (005). [1] J. Yepez, G. Vahala, L. Vahala, an M. Soe, Phys. Rev. Lett. 103, (009). [13] H. N. V. Temperley an E. H. Lieb, Proc. R. Soc. A 3, 51 (1971). [14] R. J. Baxter, Exactly Solve Moels in Statistical Mechanics (Acaemic Press, Lonon, 198). [15] D. Levy, Phys. Rev. Lett. 64, 499 (1990). [16] D. Aharonov, V. Jones, an Z. Lanau, in STOC 06: Proceeings of the 38th ACM Symposium on Theory of Computing (ACM, New York, NY, USA, 006), p. 47. [17] L. H. Kauffman an S. J. Lomonaco, Jr., in Quantum Information an Computation V, eitebye.j.donkor,a.r.pirich,an H. E. Brant (SPIE, 007, Orlano, FL, USA, 007) Vol. 6573, p T. [18] V. Jones, Bull. New Ser. Am. Math. Soc. 1, 103 (1985). [19] P. Joran an E. Wigner, Z. Phys. A 47, 631 (198). [0] A. L. Fetter an J. D. Walecka, Quantum Theory of Many- Particle Systems (McGraw-Hill, New York, 1971). [1] A. Einstein, B. Poolsky, an N. Rosen, Phys. Rev. 47, 777 (1935). [] J. Yepez, in Quantum Information an Computation VII, eite by E. J. Donkor, A. R. Pirich, an H. E. Brant (SPIE, Orlano, Fl, USA, 009), Vol. 734, p. 7340R. [3] D. M. Stamper-Kurn, M. R. Anrews, A. P. Chikkatur, S. Inouye, H. J. Miesner, J. Stenger, an W. Ketterle, Phys. Rev. Lett. 80, 07 (1998). [4] T.-L. Ho, Phys. Rev. Lett. 81, 74 (1998). [5] A. Fetter an A. Svizinsky, J. Phys. Conens. Matter 13, 135 (001). [6] J. Yepez, Phys. Rev. E 63, (001). [7] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, an H. Weinfurter, Phys. Rev. A 5, 3457 (1995)