A Geometric Algebra Perspective On Quantum Computational Gates And Universality In Quantum Computing
aa r X i v : . [ m a t h - ph ] J un A Geometric Algebra Perspective On Quantum Computational Gates AndUniversality In Quantum Computing
Carlo Cafaro ∗ and Stefano Mancini † Dipartimento di Fisica, Universit`a di Camerino, I-62032 Camerino, Italy
We investigate the utility of geometric (Clifford) algebras (GA) methods in two specific applica-tions to quantum information science. First, using the multiparticle spacetime algebra (MSTA, thegeometric algebra of a relativistic configuration space), we present an explicit algebraic descriptionof one and two-qubit quantum states together with a MSTA characterization of one and two-qubitquantum computational gates. Second, using the above mentioned characterization and the GA de-scription of the Lie algebras SO (3) and SU (2) based on the rotor group Spin + (3, 0) formalism, wereexamine Boykin’s proof of universality of quantum gates. We conclude that the MSTA approachdoes lead to a useful conceptual unification where the complex qubit space and the complex spaceof unitary operators acting on them become united, with both being made just by multivectors in real space. Finally, the GA approach to rotations based on the rotor group does bring conceptualand computational advantages compared to standard vectorial and matricial approaches. PACS numbers: quantum information (03.67.-a); quantum gates (03.67. Lx); geometric algebra (02.10.-v).
I. INTRODUCTION
Geometric (Clifford) algebra (GA) [1, 2] is a universal language for physics based on the mathematics of Cliffordalgebra. Applications of GA in physics span from quantum theory and gravity [3, 4] to classical electrodynamics [5]reaching even massive classical electrodynamics with Dirac’s magnetic monopoles [6, 7].There are some natural ways of including Clifford algebra and GA in quantum information science (QIS) motivatedby physical reasons [8, 9]. For instance, any qubit (quantum bit, elementary carrier of quantum information) modelledas a spin- system can be regarded as a 2 × × × formal reformulations of some of the most importantoperations of quantum computing in the multiparticle geometric algebra formalism have been presented [10]. Ina less conventional and very recent GA approach to quantum computing, the possibility of performing quantum-like algorithms using GA structures without involving quantum mechanics has been explored [11–13]. Within thisapproach, the standard tensor product is replaced by the geometric product and entangled states are replaced bymultivectors with a geometrical interpretation in terms of ” bags of shapes ”. Such GA approach brings new conceptualelements in QIS and the formalism of quantum computation loses its microscopic flavor when viewed from this novelGA point of view. Indeed, non-microworld implementations of quantum computing are suggested since there is nofundamental reason to believe that quantum computation has to be associated with physical systems described byquantum mechanics [13].In [9], an extended discussion on applications of GA techniques in quantum information is presented, however thefundamental concept of universality in quantum computing is not discussed. In [10], the GA formulation of the Toffoliand Fredkin three-qubit quantum gates is introduced but no explicit characterization of all one and two-qubit quantumgates appears. Here, inspired by these two works, we present a (complementary and self-contained) compact, explicitand expository GA characterization of one and two qubit quantum gates together with a novel GA-based perspectiveon the concept of universality in quantum computation. First, we present an explicit multiparticle spacetime algebra(MSTA, the geometric Clifford algebra of a relativistic configuration space) [14–17] algebraic description of one andtwo-qubit quantum states (for instance, the 2-qubit Bell states) together with a MSTA characterization of one (bit-flip,phase-flip, combined bit and phase flip quantum gates, Hadamard gate, rotation gate, phase gate and π -gate) andtwo-qubit quantum computational gates (CNOT, controlled-phase and SWAP quantum gates) [18]. Second, usingthe above mentioned explicit characterization and the GA description of the Lie algebras SO (3) and SU (2) basedon the rotor group Spin + (3, 0) formalism, we reexamine Boykin’s proof of universality of quantum gates [19, 20]. Weconclude that the MSTA approach does lead to a useful conceptual unification where the complex qubit space and ∗ Electronic address: [email protected] † Electronic address: [email protected] the complex space of unitary operators acting on them become united, with both being made just by multivectorsin real space. Furthermore, the GA approach to rotations based on the rotor group clearly brings conceptual andcomputational advantages compared to standard vectorial and matricial approaches.The layout of this article is as follows. In Section II, the basic MSTA formalism for the GA characterization ofelementary gates for quantum computation is presented. In Section III, we present an explicit GA characterization ofone and two-qubit quantum states together with a GA characterization of one and two-qubit quantum computationalgates. Furthermore, we briefly discuss the extension of the MSTA formalism to density matrices. In Section IV, usingthe above mentioned explicit characterization and the GA description of the Lie algebras SO (3) and SU (2) basedon the rotor group Spin + (3, 0) formalism, we reexamine Boykin’s proof of universality of quantum gates [19, 20].Finally, our final remarks are presented in Section V. II. MULTIPARTICLE SPACETIME ALGEBRA
In this Section, we present the basic MSTA formalism for the GA characterization of elementary gates for quantumcomputation.
A. The n -Qubit Spacetime Algebra Formalism It is commonly believed that complex space notions and an imaginary unit i C are fundamental in quantum mechan-ics. However using spacetime algebra (STA, the geometric Clifford algebra of real 4-dimensional Minkowski spacetime,[2]) it has been shown how the i C appearing in the Dirac, Pauli and Schrodinger equations has a geometrical inter-pretation in terms of rotations in real spacetime [21]. This becomes clear once introduced the geometric algebra of arelativistic configuration space, the so-called multiparticle spacetime algebra (MSTA) [14–17].In the orthodox formulation of quantum mechanics, the tensor product is used in constructing both multiparticlestates and many of the operators acting on these states. It is a notational device for explicitly isolating the Hilbertspaces of different particles. Geometric algebra attempts to justify from a foundational point of view the use of thetensor product in nonrelativistic quantum mechanics in terms of the underlying geometry of space-time [15]. The GAformalism provides an alternative representation of the tensor product in terms of the geometric product . Motivatedby the usefulness of the STA formalism in describing a single-particle quantum mechanics, the MSTA approach tomultiparticle quantum mechanics in both non-relativistic and relativistic settings was originally [15] introduced withthe hope that it would also provide both computational and, above all, interpretational advances in multiparticlequantum theory. Conceptual advances are expected to arise by exploiting the special geometric insights that theMSTA approach provides. The unique feature of the MSTA is that it implies a separate copy of the time dimensionfor each particle, as well as the three spatial dimensions. It constitutes an attempt to construct a solid conceptualframework for a multi-time approach to quantum theory. Therefore, the main original motivation for using suchformalism is the possibility of shedding light on issues of locality and causality in quantum theory. Indeed, interestingapplications of the MSTA method devoted to the reexamination of Holland’s causal interpretation of a system oftwo spin- particles [22] appear in [16, 17]. Following this line of investigation, in this article we apply the MSTAmethod to express qubits and quantum gates and to revisit in geometric algebra terms Boykin’s proof of universalityin quantum computing.The multiparticle spacetime algebra provides the ideal algebraic structure for studying multiparticle states andoperators. MSTA is the geometric algebra of n -particle configuration space which, for relativistic systems, consistsof n copies (each copy is a 1-particle space) of Minkowski spacetime. A suitable basis for the MSTA is given by theset (cid:8) γ aµ (cid:9) , where µ = 0,.., 3 labels the spacetime vector and a = 1,.., n labels the particle space. These basis vectorssatisfy the orthogonality conditions γ aµ · γ bν = δ ab η µν with η µν = diag(+, − , − , − ). Vectors from different particlespaces anticommute as a consequence of their orthogonality. Note that a basis for the entire MSTA has 2 n degreesof freedom, dim R [ cl (1, 3)] n = 2 n . In nonrelativistic quantum mechanics, all of the individual time coordinates areidentified with a single absolute time. We take this vector to be γ a for each a . Spatial vectors relative to thesetimelike vectors are modeled as bivectors through a spacetime split. A basis set of relative vectors is then defined by σ ak def = γ ak γ a , with k = 1,.., 3 and a = 1,.., n . For each particle space the set { σ ak } generates the geometric algebra ofrelative space cl (3) ∼ = cl + (1, 3). Each particle space has a basis given by,1, { σ k } , { iσ k } , i , (1)where the volume element i (the pseudoscalar , the highest grade multivector) is defined by i def = σ σ σ (suppressingthe particle space indices). The basis in (1) defines the Pauli algebra (the geometric algebra of the 3-dimensionalEuclidean space, [2]) but in GA the three Pauli σ k are no longer viewed as three matrix-valued components of a singleisospace vector, but as three independent basis vectors for real space. Notice that unlike spacetime basis vectors,relative vectors { σ ak } from separate particle spaces commute, σ ak σ bj = σ bj σ ak , a = b . It turns out that the { σ ak } generatethe direct product space [ cl (3)] n def = cl (3) ⊗ ... ⊗ cl (3) of n copies of the geometric algebra of the 3-dimensional Euclideanspace. Within the MSTA formalism, Pauli spinors (a spin- quantum system is an adequate model of quantum bit)may be represented as elements of the even subalgebra of the Pauli algebra spanned by { iσ k } and isomorphic tothe quaternion algebra. This space is a 4-dimensional real space where a general even element can be written as, ψ = a + a k iσ k , where a and a k with k = 1, 2, 3 are real scalars. An ordinary quantum state contains a pair of complex numbers, α and β , | ψ i = (cid:18) αβ (cid:19) = (cid:18) Re α + i C Im α Re β + i C Im β (cid:19) . (2)In [14], it was established a 1 ↔ | ψ i = (cid:18) a + i C a − a + i C a (cid:19) ↔ ψ = a + a k iσ k . (3)where the coefficients a and a k ∈ R . Multivectors { iσ , iσ , iσ } are the computational basis states of the real4-dimensional even subalgebra corresponding to the two-dimensional Hilbert space H with standard computationalbasis given by B H def = {| i , | i} . In the GA formalism, | i ↔ ψ (GA) | i def = 1, | i ↔ ψ (GA) | i def = − iσ . (4)The action of the conventional quantum Pauli operators n ˆΣ k , i C ˆ I o translates as [14],ˆΣ k | ψ i ↔ σ k ψσ , (5)with k = 1, 2, 3 and, i C | ψ i ↔ ψiσ . (6)In synthesis, in the single-particle theory, non-relativistic states are constructed from the even subalgebra of thePauli algebra with a basis provided by the set { iσ k } with k = 1, 2, 3. The role of the (single) imaginary unit ofconventional quantum theory is played by right multiplication by iσ . Verifying that this translation scheme worksproperly is just a matter of simple computations. Indeed, from (3) and (5) we obtain,ˆΣ | ψ i = (cid:18) − a + i C a a + i C a (cid:19) ↔ − a + a iσ − a iσ + a iσ = σ (cid:0) a + a k iσ k (cid:1) σ ,ˆΣ | ψ i = (cid:18) a + i C a − a + i C a (cid:19) ↔ a + a iσ + a iσ + a iσ = σ (cid:0) a + a k iσ k (cid:1) σ ,ˆΣ | ψ i = (cid:18) a + i C a a − i C a (cid:19) ↔ a − a iσ − a iσ + a iσ = σ (cid:0) a + a k iσ k (cid:1) σ . (7)In the n -particle algebra there will be n -copies of iσ , namely iσ a with a = 1,.., n . However, in order to faithfullymirror conventional quantum mechanics, the right-multiplication by all of these must yield the same result. Therefore,it must be ψiσ = ψiσ = ... = ψiσ n − = ψiσ n . (8)Relations in (8) are obtained by introducing the n -particle correlator E n defined as, E n def = n Y b =2 (cid:0) − iσ iσ b (cid:1) , (9)and satisfying E n iσ a = E n iσ b = J n ; ∀ a , b . Notice that E n in (9) has been defined by picking out the a = 1 space andcorrelating all the other spaces to this. However, the value of E n is independent of which of the n spaces is chosen andcorrelated to. The complex structure is defined by J n = E n iσ a with J n = − E n . Right-multiplication by the quantumcorrelator E n is a projection operation that reduces the number of real degrees of freedom from 4 n = dim R (cid:2) cl + (3) (cid:3) n to the expected 2 n +1 = dim R H n . The projection can be interpreted physically as locking the phases of the variousparticles together. The reduced even subalgebra space will be denoted by (cid:2) cl + (3) (cid:3) n /E n . Multivectors belongingto this space can be regarded as n -particle spinors (or, n -qubit states), analogous to cl + (3) for a single particle. Insynthesis, the extension to multiparticle systems involves a separate copy of the STA for each particle and the standardimaginary unit induces correlations between these particle spaces. B. An Example: The -Qubit Spacetime Algebra Formalism Quantum theory works with a single imaginary unit i C , but in the 2-particle algebra there are two bivectors playingthe role of i C , iσ and iσ . Right-multiplication of a state by either of these has to result in the same state in orderfor the GA treatment to faithfully mirror standard quantum mechanics. Therefore it must be, ψiσ = ψiσ . (10)Manipulation of (10) yields ψ = ψE , where, E def = 12 (cid:0) − iσ iσ (cid:1) , E = E . (11)Right-multiplication by E is a projection operation. If we include this factor on the right of all states, the numberof real degrees of freedom decrease from 16 to the expected 8. The multivectorial basis B cl + (3) ⊗ cl + (3) spanning the16-dimensional geometric algebra cl + (3) ⊗ cl + (3) is given by, B cl + (3) ⊗ cl + (3) def = (cid:8) iσ l , iσ k , iσ l iσ k (cid:9) , (12)with k and l = 1, 2, 3. Right-multiplying the multivectors in B cl + (3) ⊗ cl + (3) by the quantum projection operator E ,we obtain, B cl + (3) ⊗ cl + (3) E → B cl + (3) ⊗ cl + (3) E def = (cid:8) E , iσ l E , iσ k E , iσ l iσ k E (cid:9) . (13)After some straightforward algebra, it follows that, E = − iσ iσ E , iσ E = − iσ iσ E , iσ E = iσ iσ E , iσ E = iσ E , iσ E = − iσ iσ E , iσ iσ E = − iσ iσ E , iσ iσ E = iσ iσ E , iσ iσ E = iσ E . (14)Therefore, a suitable basis for the 8-dimensional reduced even subalgebra (cid:2) cl + (3) ⊗ cl + (3) (cid:3) /E is given by, B [ cl + (3) ⊗ cl + (3) ] /E def = (cid:8) iσ , iσ , iσ , iσ , iσ iσ , iσ iσ , iσ iσ (cid:9) . (15)The basis in (15) spans (cid:2) cl + (3) ⊗ cl + (3) (cid:3) /E and is the analog of a suitable standard complex basis spanning thecomplex Hilbert space H . The spacetime algebra representation of a direct-product 2-particle Pauli spinor (a 2-qubits quantum state) is now given by ψ φ E , where ψ and φ are spinors (even multivectors) in their own spaces, | ψ , φ i ↔ ψ φ E . A GA version of a standard complete basis for 2-particle spin states is provided by, | i ⊗ | i = (cid:18) (cid:19) ⊗ (cid:18) (cid:19) ↔ E , | i ⊗ | i = (cid:18) (cid:19) ⊗ (cid:18) (cid:19) ↔ − iσ E , | i ⊗ | i = (cid:18) (cid:19) ⊗ (cid:18) (cid:19) ↔ − iσ E , | i ⊗ | i = (cid:18) (cid:19) ⊗ (cid:18) (cid:19) ↔ iσ iσ E . (16)For instance, a standard entangled state between a pair of 2-level systems, a spin singlet state is defined as, | ψ singlet i def = 1 √ (cid:26)(cid:18) (cid:19) ⊗ (cid:18) (cid:19) − (cid:18) (cid:19) ⊗ (cid:18) (cid:19)(cid:27) = 1 √ | i − | i ) . (17)From (11), (16) and (17), it follows that the GA analog of | ψ singlet i is given by, H ∋ | ψ singlet i ↔ ψ (GA)singlet ∈ (cid:2) cl + (3) (cid:3) , (18)where, ψ (GA)singlet = 12 (cid:0) iσ − iσ (cid:1) (cid:0) − iσ iσ (cid:1) . (19)Furthermore, the role of multiplication by the quantum imaginary i C for 2-particle states is taken by right-sidedmultiplication by J , J = Eiσ = Eiσ = 12 (cid:0) iσ + iσ (cid:1) , (20)so that J = − E . The action of 2-particle Pauli operators is the following,ˆΣ k ⊗ ˆ I | ψ i ↔ − iσ k ψJ , ˆΣ k ⊗ ˆΣ l | ψ i ↔ − iσ k iσ l ψE , ˆ I ⊗ ˆΣ k | ψ i ↔ − iσ k ψJ . (21)For instance, the second Equation in (21) follows from the following line of reasoning,ˆΣ l | ψ i ↔ σ l ψσ = σ l ψEσ = − σ l ψEiiσ = − iσ l ψEiσ = − iσ l ψJ , (22)and therefore, ˆΣ k ⊗ ˆΣ l | ψ i ↔ (cid:0) − iσ k (cid:1) (cid:0) − iσ l (cid:1) ψJ = − iσ k iσ l ψE . (23)Finally, recalling that i C ˆΣ k | ψ i ↔ iσ k ψ , we point out that, i C ˆΣ k ⊗ ˆ I | ψ i ↔ iσ k ψ and, ˆ I ⊗ i C ˆΣ k | ψ i ↔ iσ k ψ . (24)More details on the MSTA formalism can be found in [14–17]. III. GEOMETRIC ALGEBRA AND QUANTUM COMPUTATION
Interesting quantum computations may require constructions of complicated computational networks with severalgates acting on n -qubits defining a non-trivial quantum algorithm. Therefore it is of great practical importancefinding a convenient universal set of quantum gates. A set of quantum gates n ˆ U i o is said to be universal if anylogical operation ˆ U L can be written as [18] , ˆ U L = Y ˆ U l ⊂ { ˆ U i } ˆ U l . (25)In this Section, we present an explicit GA characterization of 1 and 2-qubit quantum states together with a GAcharacterization of a universal set of quantum gates for quantum computation. Furthermore, we mention the extensionof the MSTA formalism to density matrices. A. Geometric Algebra and 1-Qubit Quantum Computing
We consider simple circuit models of quantum computation with 1-qubit quantum gates in the GA formalism.
Quantum NOT Gate (or Bit Flip Quantum Gate) . A nontrivial reversible operation we can apply to a single qubitis the NOT operation (gate) denoted by the symbol ˆΣ . For the sake of simplicity, we will first study the action ofquantum gates in the GA formalism acting on 1-qubit quantum gates given by ψ (GA) | q i = a + a iσ . Then, ˆΣ (GA)1 isdefined as, ˆΣ | q i def = | q ⊕ i ↔ ψ (GA) | q ⊕ i def = σ (cid:0) a + a iσ (cid:1) σ . (26)Recalling that the unit pseudoscalar i def = σ σ σ is such that iσ k = σ k i with k = 1, 2, 3 and recalling the geometricproduct rule, σ i σ j = σ i · σ j + σ i ∧ σ j = δ ij + iε ijk σ k , (27)we obtain, ˆΣ | q i def = | q ⊕ i ↔ ψ (GA) | q ⊕ i = − (cid:0) a + a iσ (cid:1) . (28)For the sake of completeness, we point out that the unitary quantum gate ˆΣ (GA)1 acts on the GA computational basisstates { iσ , iσ , iσ } as follows,ˆΣ (GA)1 : 1 → − iσ , ˆΣ (GA)1 : iσ → iσ , ˆΣ (GA)1 : iσ → −
1, ˆΣ (GA)1 : iσ → iσ . (29) Phase Flip Quantum Gate.
Another nontrivial reversible operation we can apply to a single qubit is the phaseflip gate denoted by the symbol ˆΣ . In the GA formalism, the action of the unitary quantum gate ˆΣ (GA)3 on themultivector ψ (GA) | q i = a + a iσ is given by,ˆΣ | q i def = ( − q | q i ↔ ψ (GA)( − q | q i def = σ (cid:0) a + a iσ (cid:1) σ . (30)From (10) and (27) it turns out that,ˆΣ | q i def = ( − q | q i ↔ ψ (GA)( − q | q i = a − a iσ . (31)Finally, the unitary quantum gate ˆΣ (GA)3 acts on the GA computational basis states { iσ , iσ , iσ } in the followingmanner, ˆΣ (GA)3 : 1 →
1, ˆΣ (GA)3 : iσ → − iσ , ˆΣ (GA)3 : iσ → − iσ , ˆΣ (GA)3 : iσ → iσ . (32) Combined Bit and Phase Flip Quantum Gates.
A suitable combination of the two reversible operations ˆΣ and ˆΣ gives rise to another nontrivial reversible operation we can apply to a single qubit. Such operation is denoted by thesymbol ˆΣ = i C ˆΣ ◦ ˆΣ . The action of ˆΣ (GA)2 on ψ (GA) | q i = a + a iσ is given by,ˆΣ | q i def = i C ( − q | q ⊕ i ↔ ψ (GA) i C ( − q | q ⊕ i def = σ (cid:0) a + a iσ (cid:1) σ . (33)From (10) and (27) it turns out that,ˆΣ | q i def = i C ( − q | q ⊕ i ↔ ψ (GA) i C ( − q | q ⊕ i = (cid:0) a − a iσ (cid:1) iσ . (34)Indeed, using (27) and the fact that iσ k = σ k i for k = 1, 2, 3, we obtain, σ (cid:0) a + a iσ (cid:1) σ = (cid:0) a − a iσ (cid:1) iσ . (35)Finally, the action of the unitary quantum gate ˆΣ (GA)2 on the GA computational basis states { iσ , iσ , iσ } is,ˆΣ (GA)2 : 1 → iσ , ˆΣ (GA)2 : iσ →
1, ˆΣ (GA)2 : iσ → iσ , ˆΣ (GA)2 : iσ → iσ . (36) Hadamard Quantum Gate.
The GA analog of the Walsh-Hadamard quantum gate ˆ H def = ˆΣ +ˆΣ √ , ˆ H (GA) acts on ψ (GA) | q i = a + a iσ as follows,ˆ H | q i def = 1 √ | q ⊕ i + ( − q | q i ] ↔ ψ (GA)ˆ H | q i def = (cid:18) σ + σ √ (cid:19) (cid:0) a + a iσ (cid:1) σ . (37)Using (28) and (31), (37) becomes,ˆ H | q i def = 1 √ | q ⊕ i + ( − q | q i ] ↔ ψ (GA)ˆ H | q i = a √ − iσ ) − a √ iσ ) . (38)Notice that GA versions of the Hadamard transformed computational states, | + i and |−i , are given by, | + i def = | i + | i√ ↔ ψ (GA) | + i = 1 − iσ √ |−i def = | i − | i√ ↔ ψ (GA) |−i = 1 + iσ √ H (GA) acts on the GA computational basis states { iσ , iσ , iσ } as follows,ˆ H (GA) : 1 → − iσ √ H (GA) : iσ → − iσ + iσ √ H (GA) : iσ → − iσ √ H (GA) : iσ → iσ + iσ √ Rotation Gate.
The action of rotation gates ˆ R (GA) θ on ψ (GA) | q i = a + a iσ is defined as,ˆ R θ | q i def = (cid:20) i C θ )2 + ( − q − exp ( i C θ )2 (cid:21) | q i ↔ ψ (GA)ˆ R θ | q i def = a + a iσ (cos θ + iσ sin θ ) . (41)The unitary quantum gate ˆ R (GA) θ acts on the GA computational basis states { iσ , iσ , iσ } as follows,ˆ R (GA) θ : 1 →
1, ˆ R (GA) θ : iσ → iσ (cos θ + iσ sin θ ) , ˆ R (GA) θ : iσ → iσ (cos θ + iσ sin θ ) , ˆ R (GA) θ : iσ → iσ . (42) Phase Quantum Gate and π -Quantum Gate. The phase gate ˆ S (GA) acts on on ψ (GA) | q i = a + a iσ as follows,ˆ S | q i def = (cid:20) i C − q − i C (cid:21) | q i ↔ ψ (GA)ˆ S | q i def = a + (cid:0) a iσ (cid:1) iσ . (43)Furthermore, the unitary quantum gate ˆ S (GA) acts on the GA computational basis states { iσ , iσ , iσ } as follows,ˆ S (GA) : 1 →
1, ˆ S (GA) : iσ → iσ , ˆ S (GA) : iσ → − iσ , ˆ S (GA) : iσ → iσ . (44)The GA analog of the π -quantum gate ˆ T is defined as,ˆ T | q i def = " (cid:0) i C π (cid:1) − q − exp (cid:0) i C π (cid:1) | q i ↔ ψ (GA)ˆ T | q i def = 1 √ (cid:0) a + a iσ (cid:1) (1 + iσ ) . (45)Finally, the unitary quantum gate ˆ T (GA) acts on the GA computational basis states { iσ , iσ , iσ } as follows,ˆ T (GA) : 1 →
1, ˆ T (GA) : iσ → iσ (1 + iσ ) √ T (GA) : iσ → iσ (1 + iσ ) √ T (GA) : iσ → iσ . (46)In conclusion, the action of some of the most relevant 1-qubit quantum gates in the GA formalism on the GAcomputational basis states { iσ , iσ , iσ } can be summarized in the following tabular form:1- Qubit States NOT Phase Flip Bit and Phase Flip Hadamard Rotation π -Gate − iσ iσ − iσ √ iσ iσ − iσ − iσ + iσ √ iσ (cos θ + iσ sin θ ) iσ iσ ) √ iσ − − iσ iσ − iσ √ iσ (cos θ + iσ sin θ ) iσ iσ ) √ iσ iσ iσ iσ iσ + iσ √ iσ iσ → iσ . (47)Therefore, in the GA approach qubits become elements of the even subalgebra, unitary quantum gates become rotorsand the conventional complex structure of quantum mechanics is controlled by the bivector iσ .Quantum gates have a geometrical interpretation when expressed in the GA formalism. Recall that in the conven-tional approach to quantum gates, an arbitrary unitary operator on a single qubit can be written as a combinationof rotations together with global phase shifts on the qubit, ˆ U = e i C α R ˆ n ( θ ) for some real numbers α and θ and a real three-dimensional unit vector ˆ n ≡ ( n , n , n ) . For instance, the Hadamard gate ˆ H acting on a single qubit has theproperties s ˆ H ˆΣ ˆ H = ˆΣ and ˆ H ˆΣ ˆ H = ˆΣ with ˆ H = ˆ I . Therefore, ˆ H can be envisioned (up to an overall phase) asa θ = π rotation about the axis ˆ n = √ (ˆ n + ˆ n ) that rotates ˆ x to ˆ z and viceversa, ˆ H = − i C R √ (ˆ n +ˆ n ) ( π ) . In GA,rotations are handled by means of rotors. The Hadamard gate, for instance, has a simple real (no use of complex numbers is needed) geometric interpretation: it is represented by a rotor ˆ H (GA) = e − i π σ σ √ describing a rotation by π about the σ + σ √ axis. It is straightforward to show that the action of the rotor ˆ H (GA) on the 1 - qubit computationalbasis states satisfies (up to an overall irrelevant phase shift) the transformation laws appearing in (47). We point outthat when the Hadamard gate is represented by a rotor for a rotation by π , ˆ H (GA)2 = −
1. Therefore, it seems thatthe gate is more accurately represented by a reflection rather than a rotation. The phase difference may be importantwhen state amplitudes transformed by the Hadamard gate are combined with the ones transformed by other gates.In [9], it was also proposed treating the Hadamard gate as a rotation but it is now recognized the problem with thisinterpretation. Similar geometric considerations could be carried out for the other 1-qubit gates [9].
B. Geometric Algebra and -Qubit Quantum Computing We consider simple circuit models of quantum computation with 2-qubit quantum gates in the GA formalism.Before doing so, we present an explicit MSTA description of quantum Bell states.
Geometric Algebra and Bell States.
We present a GA characterization of the set of maximally entangled 2-qubitsBell states. Bell states are an important example of maximally entangled quantum states and form an orthonormalbasis B Bell in the product Hilbert space C ⊗ C ∼ = C . Consider the 2-qubit computational basis B computational = {| i , | i , | i , | i} , then the four Bell states can be constructed as follows [18], | i ⊗ | i → | ψ Bell i def = h ˆ U CNOT ◦ (cid:16) ˆ H ⊗ ˆ I (cid:17)i ( | i ⊗ | i ) = 1 √ | i ⊗ | i + | i ⊗ | i ) , | i ⊗ | i → | ψ Bell i def = h ˆ U CNOT ◦ (cid:16) ˆ H ⊗ ˆ I (cid:17)i ( | i ⊗ | i ) = 1 √ | i ⊗ | i + | i ⊗ | i ) , | i ⊗ | i → | ψ Bell i def = h ˆ U CNOT ◦ (cid:16) ˆ H ⊗ ˆ I (cid:17)i ( | i ⊗ | i ) = 1 √ | i ⊗ | i − | i ⊗ | i ) , | i ⊗ | i → | ψ Bell i def = h ˆ U CNOT ◦ (cid:16) ˆ H ⊗ ˆ I (cid:17)i ( | i ⊗ | i ) = 1 √ | i ⊗ | i − | i ⊗ | i ) . (48)In (48), ˆ H is the Hadamard gate and ˆ U CNOT is the CNOT gate. The Bell basis in C ⊗ C ∼ = C is given by, B Bell def = {| ψ Bell i , | ψ Bell i , | ψ Bell i , | ψ Bell i} , (49)where from (48) we get, | ψ Bell i = 1 √ , | ψ Bell i = 1 √ , | ψ Bell i = 1 √ − , | ψ Bell i = 1 √ − . (50)From (16) and (48), it follows that the GA version of Bell states is given by, | ψ Bell i ↔ ψ (GA)Bell = 12 (cid:0) iσ iσ (cid:1) (cid:0) − iσ iσ (cid:1) , | ψ Bell i ↔ ψ (GA)Bell = − (cid:0) iσ + iσ (cid:1) (cid:0) − iσ iσ (cid:1) , | ψ Bell i ↔ ψ (GA)Bell = 12 (cid:0) − iσ iσ (cid:1) (cid:0) − iσ iσ (cid:1) , | ψ Bell i ↔ ψ (GA)Bell = 12 (cid:0) iσ − iσ (cid:1) (cid:0) − iσ iσ (cid:1) . (51)We point out that within the MSTA formalism, there is no need either for an abstract spin space (the complex Hilbertspace H n ), containing objects which have to be operated on by quantum unitary operators (for instance, in the Bellstates example, such operators are the CNOT gates), or for an abstract index convention. The requirement for anexplicit matrix representation is also avoided. Within the MSTA formalism, the role of operators is taken over byright or left multiplication by elements from the same geometric algebra as spinors (qubits) are taken from. This is anadditional manifestation of a conceptual unification provided by GA - ”spin (or qubit) space” and ”unitary operatorsupon spin space” become united, with both being just multivectors in real space. Indeed, such a manifestation isnot a special feature of our work since it appears in most geometric algebra applications to classical and quantummathematical physics. CNOT Quantum Gate.
An convenient way to describe the CNOT quantum gate is the following [18],ˆ U = 12 h(cid:16) ˆ I + ˆΣ (cid:17) ⊗ ˆ I + (cid:16) ˆ I − ˆΣ (cid:17) ⊗ ˆΣ i , (52)where ˆ U is the CNOT gate from qubit 1 to qubit 2. It then follows that,ˆ U | ψ i = 12 (cid:16) ˆ I ⊗ ˆ I + ˆΣ ⊗ ˆ I + ˆ I ⊗ ˆΣ − ˆΣ ⊗ ˆΣ (cid:17) | ψ i . (53)From (21) and (53), we obtainˆ I ⊗ ˆ I | ψ i ↔ ψ , ˆΣ ⊗ ˆ I | ψ i ↔ − iσ ψJ , ˆ I ⊗ ˆΣ | ψ i ↔ − iσ ψJ , − ˆΣ ⊗ ˆΣ | ψ i ↔ iσ iσ ψE . (54)Finally, from (53) and (54), we get the GA version of the CNOT gate,ˆ U | ψ i ↔ (cid:0) ψ − iσ ψJ − iσ ψJ + iσ iσ ψE (cid:1) . (55) Controlled-Phase Gate.
The action of ˆ U on | ψ i ∈ H is given by [18],ˆ U | ψ i = 12 h ˆ I ⊗ ˆ I + ˆΣ ⊗ ˆ I + ˆ I ⊗ ˆΣ − ˆΣ ⊗ ˆΣ i | ψ i . (56)From (21) and (56), we obtainˆ I ⊗ ˆ I | ψ i ↔ ψ , ˆΣ ⊗ ˆ I | ψ i ↔ − iσ ψJ , ˆ I ⊗ ˆΣ | ψ i ↔ − iσ ψJ , − ˆΣ ⊗ ˆΣ | ψ i ↔ iσ iσ ψE . (57)Finally, from (56) and (57), we obtain the GA version of the controlled-phase quantum gate,ˆ U | ψ i ↔ (cid:0) ψ − iσ ψJ − iσ ψJ + iσ iσ ψE (cid:1) . (58) SWAP Gate.
The action of ˆ U on | ψ i ∈ H is given by [18],ˆ U | ψ i = 12 (cid:16) ˆ I ⊗ ˆ I + ˆΣ ⊗ ˆΣ + ˆΣ ⊗ ˆΣ + ˆΣ ⊗ ˆΣ (cid:17) | ψ i . (59)From (21) and (59), we obtainˆ I ⊗ ˆ I | ψ i ↔ ψ , ˆΣ ⊗ ˆΣ | ψ i ↔ − iσ iσ ψE , ˆΣ ⊗ ˆΣ | ψ i ↔ − iσ iσ ψE , ˆΣ ⊗ ˆΣ | ψ i ↔ − iσ iσ ψE . (60)Finally, from (59) and (60), we obtain the GA version of the SWAP gate,ˆ U | ψ i ↔ (cid:0) ψ − iσ iσ ψE − iσ iσ ψE − iσ iσ ψE (cid:1) . (61)In conclusion, the action of some of the most relevant 2-qubit quantum gates in the GA formalism on the GAcomputational basis B [ cl + (3) ⊗ cl + (3) ] /E can be summarized in the following tabular form:2- Qubit Gates Qubit States GA Action of Gates on States
CNOT ψ (cid:0) ψ − iσ ψJ − iσ ψJ + iσ iσ ψE (cid:1) Controlled-Phase Gate ψ (cid:0) ψ − iσ ψJ − iσ ψJ + iσ iσ ψE (cid:1) SWAP ψ (cid:0) ψ − iσ iσ ψE − iσ iσ ψE − iσ iσ ψE (cid:1) . (62)Two-qubit quantum gates have a geometric interpretation in terms of rotations as well. For instance, the CNOT gatedescribes a rotation in one qubit space conditional on the state of another qubit it is correlated with. The generalexpression of the corresponding operator in GA is given by (cid:16) ˆ U (cid:17) (GA) = e − i π σ ( − σ ) . This operator rotatesthe first qubit by π about the axis σ in those 2-qubit states where the second qubit is along the − σ axis. Similargeometric considerations could be considered for the other 2-qubit gates [10].0 C. Geometric Algebra and Density Operators
For the sake of completeness, we point out that statistical aspects of quantum systems cannot be described in termsof a single wavefunction. Instead, they can be properly handled in terms of density matrices. The density matrix fora pure state is given by, ˆ ρ pure = | ψ i h ψ | = αα ∗ αβ ∗ βα ∗ ββ ∗ ! . (63)The expectation value of any observable ˆ O associated with the state | ψ i can be obtained from ˆ ρ pure by writing D ψ | ˆ O | ψ E =Tr (cid:16) ˆ ρ pure ˆ O (cid:17) . The GA version of ˆ ρ pure is,ˆ ρ pure → ρ (GA)pure = ψ
12 (1 + σ ) ψ † = 12 (1 + s ) , (64)where s def = ψσ ψ † is the spin vector [16]. From a geometric point of view, ρ (GA)pure is just the sum of a scalar and avector. The density matrix for a mixed state ˆ ρ mixed is the weighted sum of the density matrices for the pure states,ˆ ρ mixed = n X j =1 ˆ ρ j = n X j =1 p j | ψ j i h ψ j | , (65)with p j ∈ R for j = 1,..., n and p +...+ p n = 1 . In the GA formalism, addition is well-defined and the geometricalgebra version of ˆ ρ mixed becomes the sum,ˆ ρ mixed → ρ (GA)mixed = 12 n X j =1 ( p j + p j s j ) = 12 (1 + P ) , (66)where P is the ensemble-average polarization vector (average spin vector) with length k P k ≤
1. The magnitude of P measures the degree of alignment among the unit length polarization vectors of the individual numbers of the ensemble.We point out that ρ (GA)mixed is the geometric algebra expression of the density operator of an ensemble of identical andnon-interacting qubits. More generally, we could also consider density operators of interacting multi-qubit systems.The MSTA version of the density matrix of n -interacting qubits reads, ρ (GA)multi-qubit = ( ψE n ) E + ( ψE n ) ∼ , (67)where E n is the n -particle correlator and E + def = E E ... E n + is the geometric product of n -idempotents with E k ± = ± σ k and k = 1,..., n . The symbol tilde denotes the space-time reverse and the over-line denotes the ensemble-average.A more detailed application of the GA formalism to general density matrices appears in [9]. IV. ON THE UNIVERSALITY OF QUANTUM GATES AND GEOMETRIC ALGEBRA
In this Section, using the above mentioned explicit characterization and the GA description of the Lie algebras SO (3) and SU (2) based on the rotor group Spin + (3, 0) formalism, we reexamine Boykin’s proof of universality ofquantum gates. In the first part, we introduce the rotor group. In the second part, we introduce few universal sets ofquantum gates. In the last part, we present our GA-based proof. A. On SO (3) , SU (2) and the Rotor Group in Geometric Algebra Since Boykin’s proof heavily relies on the properties of rotations in three-dimensional space and on the localisomorphism between SO (3) and SU (2), we briefly present the the GA description of such Lie groups via the rotorgroup Spin + (3, 0).1
1. Remarks on SO (3) and SU (2) Two important groups in physics are the 3-dimensional Lie groups SO (3) with Lie algebra so (3) and SU (2) withLie algebra su (2) [23]. The former is the group of rotations of three-dimensional space, i.e. the group of orthogonaltransformations with determinant 1, SO (3) def = (cid:8) M ∈ GL (3, R ) : M M t = M t M = I × , det M = 1 (cid:9) , (68)where ” t ” denotes the transpose of a matrix and GL (3, R ) is the set of non-singular linear transformations in R which are represented by 3 × × SU (2) def = (cid:8) M ∈ GL (2, C ) : M M † = M † M = I × , det M = 1 (cid:9) , (69)where ” † ” denotes the Hermitian conjugate and GL (2, C ) is the set of non-singular linear transformations in C whichare represented by 2 × so (3) and su (2) are isomorphic, so (3) ∼ = su (2). The Lie groups SO (3) and SU (2) are locally isomorphic, they are indistinguishable at the level ofinfinitesimal transformations. However, they differ at a global level, i. e. far from identity. This means that SO (3)and SU (2) are not isomorphic. In SO (3) a rotation by 2 π is the same as the identity. Instead, SU (2) is periodic onlyunder rotations by 4 π . This means that an object that picks a minus sign under a rotation by 2 π is an acceptablerepresentation of SU (2), while it is not an acceptable representation of SO (3). Spin particles or qubits need tobe rotated 720 in order to come back to the same state [24]. Topologically, SU (2) is the 3-sphere S , SU (2) ≈ S .Instead, SO (3) is topologically equivalent to the projective space R P where R P results from S by identifying pairsof antipodal points. This leads to conclude the actual isomorphism between groups is SU (2) / Z ∼ = SO (3). In formalmathematical terms, there is a not faithful representation κ of SU (2) as a group of rotations of R , κ : SU (2) ∋ U SU (2) (cid:16) ~A , θ (cid:17) def = exp ~ Σ2 i C · ~Aθ ! R SO (3) (cid:16) ~A , θ (cid:17) def = exp (cid:16) ~E · ~Aθ (cid:17) ∈ SO (3), (70)for any vector ~A = ( A , A , A ). For the sake of mathematical correctness, we point out that the use of the dot-notation in (70) (and in the following equations (72), (83), (106)) is indeed an abuse of notation for the Euclidean innerproduct because ~A is geometrically just a vector in R while ~ Σ are operators (Pauli matrices) on a two-dimensionalHilbert space. The vector ~E = ( E , E , E ) forms a basis of infinitesimal generators of the Lie algebra so (3) of SO (3), E = −
10 1 0 , E = − , E = − , (71)and they satisfy the following commutation relations, [ E l , E m ] = ε lmk E k . The infinitesimal generators of the Liealgebra su (2) of SU (2) are i C ~ Σ = ( i C Σ , i C Σ , i C Σ ) satisfying the commutation relations, [Σ l , Σ m ] = 2 i C ε lmk Σ k .This commutations relations are the same as for SO (3) if one uses Σ l i C as the new basis for su (2). The map κ isexactly 2 : 1 and therefore to a rotation of R about an axis given by a unit vector ~A through an angle θ radians oneassociates two 2 × ~ Σ2 i C · ~Aθ ! , exp ~ Σ2 i C · ~A ( θ + 2 π ) ! . (72)In other words, SO (3) not only has the usual representation by 3 × × C . The complex vectors (cid:16) ψ ψ (cid:17) t ∈ C on which SO (3) acts in this double-valuedway are called spinors. In mathematical terms, SU (2) furnishes naturally a spinor representation of the 2-fold coverof SO (3). When SU (2) is thought of as the 2-fold cover of SO (3), it is called the spin group Spin (3). It is remarkablypowerful to represent three-dimensional rotations in terms of two-dimensional unitary transformations. In quantuminformation science, this is especially true in proving certain circuit identities, in the characterization of the general1-qubit state and in the construction of the Hardy state [25]. Indeed, this representation plays also a key role in theproof of universality of quantum gates provided by Boykin et al . For instance, the surjective homeomorphism κ is apowerful tool for investigating the product of two or more rotations. This is a consequence of the fact that the Paulimatrices satisfy very simple product rules, Σ l Σ m = δ lm + i C ε lmk Σ k . The infinitesimal generators { E l } with l = 1, 2,3 of so (3) do not satisfy such simple product relations. For instance, E = diag (0, − −
2. Remarks on the rotor group
One of the most powerful applications of geometric algebra is to rotations. Within GA, rotations are handledthrough the use of rotors . Rotors also provide a convenient framework for studying Lie groups and Lie algebras. Letus introduce few definitions. More details on the mathematical structure of Clifford algebras appears in [26]Let G ( p , q ) denote the GA of a space of signature p , q with p + q = n and let V be the space of grade-1 multivectors.Then, the pin group (with respect to the geometric product) Pin ( p , q ) is defined as, Pin ( p , q ) def = (cid:8) M ∈ G ( p , q ) : M aM − ∈ V ∀ a ∈ V , M M † = ± (cid:9) , (73)where ” † ” denotes the reversion operation in GA (for instance, ( a a ) † = a a ). The elements of the pin group splitinto even-grade and odd-grade elements. The even-grade multivectors { S } of the pin group form a subgroup calledthe spin group Spin ( p , q ), Spin ( p , q ) def = (cid:8) S ∈ G + ( p , q ) : SaS − ∈ V ∀ a ∈ V , SS † = ± (cid:9) , (74)where G + ( p , q ) denotes the even subalgebra of G ( p , q ). Finally, rotors are elements { R } of the spin group satisfyingthe further constraint that RR † = +1. These elements define the so-called rotor group Spin + ( p , q ), Spin + ( p , q ) def = (cid:8) R ∈ G + ( p , q ) : RaR † ∈ V ∀ a ∈ V , RR † = +1 (cid:9) . (75)For Euclidean spaces, Spin ( n , 0) = Spin + ( n , 0). Therefore, for such spaces, there is no distinction between the spingroup and the rotor group.In GA, the rotation of a vector a through θ in the plane generated by two unit vectors m and n is defined by thedouble-sided half-angle transformation law, a → a ′ def = RaR † . (76)The rotor R is defined by, R def = nm = n · m + n ∧ m ≡ exp (cid:18) − B θ (cid:19) , (77)where the bivector B is such that, B = m ∧ n sin θ , B = −
1. (78)Rotors provide a way of handling rotations that is unique to GA and this is a consequence of the definition of thegeometric product. Notice that rotors are geometric products of two unit vectors and therefore they are mixed-gradeobjects. The rotor R on its own has no significance, which is to say that no meaning should be attached to theseparate scalar and bivector terms. When R is written as the exponential of the bivector B (all rotors near the origincan be written as the exponential of a bivector and the exponential of a bivector always returns to a rotor), however,the bivector has a clear geometric significance, as does the vector formed from RaR † . This illustrates a central featureof GA, which is that both geometrically meaningful objects (vectors, planes, etc.) and the elements (operators) thatact on them (in this case, rotors or bivectors) are contained in the same geometric Clifford algebra. Notice that R and − R generate the same rotation, so there is a two-to-one map between rotors and rotations. Formally, the rotorgroup provides a double-cover representation of the rotation group.The Lie algebra of the rotor group Spin + (3, 0) is defined in terms of the bivector algebra,[ B l , B m ] = 2 B l × B m = − ε lmk B k , (79)where ” × ” denotes the commutator product of two multivectors in GA and, B = σ σ = iσ , B = σ σ = iσ , B = σ σ = iσ . (80)The commutator of a bivector with a second bivector produces a third bivector. That is, the space of bivectors isclosed under the commutator product. This closed algebra defines the Lie algebra of the associated rotor group. Thegroup is formed by the act of exponentiation. Furthermore, notice that the product of bivectors satisfies, B l B m = − δ lm − ε lmk B k . (81)3The symmetric part of this product is a scalar, whereas the antisymmetric part is a bivector. As a final remark, wepoint out that the algebra of bivectors is similar to the algebra of the generators of the quaternions. Thus, quaternionscan be identified with bivectors within the GA approach. The relations among SO (3), SU (2) and the rotor group aresummarized as follows, Lie Groups Lie Algebras Product Rules Operator , Vectors SO (3) [ E l , E m ] = ε lmk E k not useful Orthogonal transformations, vectors in R SU (2) [Σ l , Σ m ] = 2 i C ε lmk Σ k Σ l Σ m = δ lm + i C ε lmk Σ k Unitary operators, spinors
Spin + (3, 0) [ B l , B m ] = − ε lmk B k B l B m = − δ lm − ε lmk B k Rotors (or, bivectors), multivectors . (82)Therefore, two central features of GA emerge: 1) the GA provides a very clear and compact method for encodingrotations which is considerably more powerful than working with matrices; 2) Both geometrically meaningful objects(vectors, planes, etc.) and the elements (operators) that act on them (in this case, rotors or bivectors) are containedin the same geometric Clifford algebra.
B. Universal Sets of Quantum Gates
Quantum computational gates are input-output devices whose inputs and outputs are discrete quantum variablessuch as spins. As a matter of fact, recall that the most general 2 × U SU (2) (ˆ n , θ ) def = e − i C θ ˆ n · ~ Σ = ˆ I cos (cid:18) θ (cid:19) − i C ˆ n · ~ Σ sin (cid:18) θ (cid:19) . (83)Therefore, we are entitled to think of a qubit as the state of a spin- object and an arbitrary unitary transfor-mation (quantum gate) acting on the state (aside from a possible rotation of the overall phase) is a rotation ofthe spin. A set of gates is adequate if any quantum computation can be performed with arbitrary precision bynetworks consisting only of replicas of gates from that set. A gate is universal if by itself it forms an adequateset, i. e. if any quantum computation can be performed by a network containing replicas of only this gate. Thefirst example of universal gate is the universal Deutsch three-bit gate [27]. In the network’s computational basis B H = {| i , | i , | i , | i , | i , | i , | i , | i} , it is given by the 8 × D (Deutsch)universal ( γ ), D (Deutsch)universal ( γ ) def = I × O × O × D × ( γ ) ! , (84)where I l × k is the l × k identity matrix, O l × k is the l × k null matrix and D × ( γ ) is defined as, D × ( γ ) def = i C cos (cid:0) πγ (cid:1) sin (cid:0) πγ (cid:1) sin (cid:0) πγ (cid:1) i C cos (cid:0) πγ (cid:1) ! . (85)The Deutsch gate depends on the parameter γ that can be any irrational number. Another important example ofuniversal quantum gate is given by the Barenco three-parameter family of universal two-bit gates [28]. In the network’scomputational basis B H = {| i , | i , | i , | i} , it is given by the 4 × A (Barenco)universal ( φ , α , θ ), A (Barenco)universal ( φ , α , θ ) def = I × O × O × A × ( φ , α , θ ) ! , (86)where I l × k is the l × k identity matrix, O l × k is the l × k null matrix and A × ( φ , α , θ ) is defined as, A × ( φ , α , θ ) def = e i C α cos θ − i C e i C ( α − φ ) sin θ − i C e i C ( α + φ ) sin θ e i C α cos θ ! . (87)The Barenco gate depends on three parameters φ , α , and θ that are fixed irrational multiples of π and of each other.More generally, it turns out that almost all two-bit (and more inputs) quantum gates are universal [29, 30].4A set of quantum gates S is said to be universal if an arbitrary unitary quantum operation can be performed witharbitrarily small error probability using a quantum circuit that only uses gates from S . An important set of logicgates in quantum computing is given by, S Clifford def = n ˆ H , ˆ P , ˆ U CNOT o . (88)The set S Clifford of Hadamard- ˆ H , phase- ˆ P and CNOT- U CNOT gates generates the so-called Clifford group, the nor-malizer N ( G n ) of the Pauli group G n in U ( n ) [31]. This set of gates is sufficient to perform fault-tolerant quantumcomputation but S Clifford is not sufficiently powerful to perform universal quantum computation. However, universalquantum computation becomes possible if the gates in the Clifford group are supplemented with the Toffoli gate [32], S (Shor)universal def = n ˆ H , ˆ P , ˆ U CNOT , ˆ U Toffoli o . (89)Shor showed that adding the Toffoli gate to the generators of the Clifford group produces the universal set S (Shor)universal .Another example of universal set of logic gates is provided by Boykin et al. in [19, 20]. The set they construct isgiven by, S (Boykin et al .)universal def = n ˆ H , ˆ P , ˆ T , ˆ U CNOT o . (90)This set is presumably easier to implement experimentally than S (Shor)universal since the ˆ T is a one-qubit gate while theToffoli gate is a three-qubit gate. C. GA reexamination of Boykin’s proof of universality
Boykin’s proof is very elegant and is solely based on the geometry of real rotations in three dimensions and on thelocal isomorphism between the Lie groups SO (3) and SU (2). In what follows, we will revisit the proof using a GAapproach based on the rotor group Spin + (3, 0) and on the algebra of bivectors, [ B l , B m ] = − ε lmk B k .The proof of universality of the basis S (Boykin et al .)universal can be presented in two steps. In the first step, it is requiredto show that Hadamard gate ˆ H and the π -phase gate ˆ T = ˆΣ form a dense set in SU (2) where,ˆΣ α = e i C πα ! , ˆΣ α | ψ i ↔ ψ (GA)ˆΣ α = σ α ψσ . (91)This means that any element ˆ U SU (2) in SU (2) can be approximated to a desired degree of precision by a finite productof ˆ H and ˆ T . In other words, when a circuit of quantum gates is used to implement some desired unitary operationˆ U , it is sufficient to have an implementation that approximates ˆ U to some specified level of accuracy. Suppose weapproximate ˆ U by some other unitary transformation ˆ U ′ . Then, the notion of the quality of an approximation of aunitary transformation can be quantified considering the so-called approximation error ε (cid:16) ˆ U , ˆ U ′ (cid:17) [33], ε (cid:16) ˆ U , ˆ U ′ (cid:17) def = max | ψ i (cid:13)(cid:13)(cid:13)(cid:16) ˆ U − ˆ U ′ (cid:17) | ψ i (cid:13)(cid:13)(cid:13) , (92)where k ψ k = p h ψ | ψ i is the Euclidean norm of | ψ i and h·|·i is the conventional inner product defined on the complexHilbert space. In the second step of the proof, it is necessary to point out that for universal computation all that isneeded is ˆ U CNOT and SU (2) [34].To show that ˆ H and ˆ T form a dense set in SU (2), the local isomorphism between SO (3) and SU (2) must beexploited. Indeed, it can be shown that using the set n ˆ H , ˆ T = ˆΣ o , we can construct quantities in this basis thatcorrespond to rotations by angles that are irrational multiples of π in SO (3, R ) about two orthogonal axes. Considerthe following two rotations in SO (3) described in terms of rotors in Spin + (3, 0), SO (3) ∋ ˆ U (1) SO (3) def = e i C λ π ˆ n · ~ Σ ↔ e in λ π ∈ Spin + (3, 0) , ˆ U (2) SO (3) def = e i C λ π ˆ n · ~ Σ ↔ e in λ π , (93)where the elements λ , λ are irrational numbers in R / Q . Let us show that rotations in (93) can be expressed interms of a suitable combination of elements in n ˆ H , ˆ T = ˆΣ o . It turns out that since SU (2) / Z ∼ = SO (3), we have Spin + (3, 0) ∋ e in λ π ↔ ˆ U (1) SU (2) def = ˆΣ − ˆΣ ∈ SU (2) and, e in λ π ↔ ˆ U (2) SU (2) def = ˆ H − ˆΣ − ˆΣ ˆ H , (94)5where ˆΣ = ˆ H ˆΣ ˆ H . Working out the details of [19, 20] and using the results presented in Section III, it turns outthat the rotor representation of ˆ U (1) SU (2) and ˆ U (2) SU (2) is given by,ˆ U (1) SU (2) ↔ R = 12 (cid:18) √ (cid:19) − √ iσ + 12 (cid:18) − √ (cid:19) iσ + 12 √ iσ , (95)and, ˆ U (2) SU (2) ↔ R = 12 (cid:18) √ (cid:19) − (cid:18) − √ (cid:19) iσ + 12 iσ + 12 (cid:18) − √ (cid:19) iσ , (96)respectively. R and R are rotors in Spin + (3, 0). Notice that, e in k λ k π = cos ( λ k π ) + n kx sin ( λ k π ) iσ + n ky sin ( λ k π ) iσ + n kz sin ( λ k π ) iσ , (97)for k = 1, 2 and unit vectors n k . Therefore, setting e in λ π = R we get,cos ( λ π ) = 12 (cid:18) √ (cid:19) , n y sin ( λ π ) = 12 (cid:18) − √ (cid:19) , n z sin ( λ π ) = 12 1 √ n x = − n z . (98)Finally, after some algebra, we obtain that λ is equal to, λ = 1 π cos − (cid:20) (cid:18) √ (cid:19)(cid:21) , (99)and the unit vector n = n x σ + n y σ + n z σ is such that,( n x , n y , n z ) = 1 r − h (cid:16) √ (cid:17)i (cid:18) − √ − √ √ (cid:19) . (100)Similarly, setting e in λ π = R , we obtain,cos ( λ π ) = 12 (cid:18) √ (cid:19) , n y sin ( λ π ) = 12 , n z sin ( λ π ) = 12 (cid:18) − √ (cid:19) , n x = − n z . (101)Finally, after some algebra, we obtain that λ = λ is equal to, λ = 1 π cos − (cid:20) (cid:18) √ (cid:19)(cid:21) , (102)and the unit vector n = n x σ + n y σ + n z σ is such that,( n x , n y , n z ) = 1 r − h (cid:16) √ (cid:17)i (cid:18) −
12 ( 12 − √ − √ (cid:19) . (103)From (100) and (103), it follows that n · n = 0. Since λ = λ ≡ λ ∈ R / Q , there exist some n ∈ N such that anyphase factor e i C φ can be approximated by e i C nλπ , e i C φ ≈ e i C nλπ , n ∈ N . (104)From (94) and (104), it follows that we have at least two dense subsets of SU (2, C ), that is to say e in α and e iβn with, α ≈ λπl (mod2 π ) and, β ≈ λπl (mod2 π ) with l ∈ N . (105)Since n and n are orthogonal vectors, we can write any element ˆ U SU (2) ∈ SU (2, C ) in the following form,ˆ U SU (2) = e i C φ ˆ n · ~ Σ ↔ e inφ = e in α e in β e in γ . (106)6Notice that the representation in (106) is analogous to Euler rotations about three orthogonal vectors. Expansion ofthe LHS of (106) leads to, e inφ = cos φ + in sin φ . (107)Expansion of the RHS of (106) yields, e in α e in β e in γ = (cos α + in sin α ) (cos β + in sin β ) (cos γ + in sin γ ) . (108)Recalling that n n = n · n + n ∧ n and that the unit vectors n and n are orthogonal, we have, n n = − n n . (109)Moreover, recalling that,sin ( α ± β ) = sin α cos β ± cos α sin β , and, cos ( α ± β ) = cos α cos β ∓ sin α sin β , (110)further expansion of (108) together with the use of (109) and (110), leads to e inφ = cos β cos ( α + γ ) + cos β sin ( α + γ ) in + sin β cos ( γ − α ) in + sin β sin ( γ − α ) n ∧ n . (111)Setting (107) equal to (111), we finally obtain cos φ = cos β cos ( α + γ ) (112)and, n sin φ = cos β sin ( α + γ ) n + sin β cos ( γ − α ) n − i sin β sin ( γ − α ) ( n ∧ n ) . (113)In conclusion, the parameters α , β and γ can be found by inverting (112) and (113) for any element in SU (2). Then,using the fact that ˆ U CNOT and SU (2) form a universal basis for quantum computing [34], the proof is completed[19, 20]. It is evident that the GA provides a very clear and compact method for encoding rotations which isconsiderably more powerful than working with matrices. Furthermore, from a conceptual point of view, a centralfeature of GA emerges as well ( although such feature appears in most geometric algebra applications ) : both vectors(grade-1 multivectors), planes (grade-2 multivectors) and the operators acting on them (in this case, rotors R orbivectors B ) are contained in the same geometric Clifford algebra. V. CONCLUSIONS AND REMARKS
In this article, we investigated the utility of GA methods in two specific applications to quantum informationscience. First, we presented an explicit multiparticle spacetime algebra description of one and two-qubit quantumstates together with a MSTA characterization of one and two-qubit quantum computational gates. Second, using theabove mentioned explicit characterization and the GA description of the Lie algebras SO (3) and SU (2) based on therotor group Spin + (3, 0) formalism, we reexamined Boykin’s proof of universality of quantum gates. We conclude thatthe MSTA approach leads to a useful conceptual unification where the complex qubit space and the complex spaceof unitary operators acting on them become united, with both being made just by multivectors in real space [35].Furthermore, the GA approach to rotations based on the rotor group clearly brings conceptual and computationaladvantages compared to standard vectorial and matricial approaches. In what follows, we present few concludingremarks.In standard quantum computation, the basic operation is the tensor product ” ⊗ ”. In the GA approach to quantumcomputing, the basic operation becomes the geometric (Clifford) product. Tensor product has no neat geometricvisualization while geometric product has clear geometric interpretations. For instance, it forms a cube ( σ σ σ ) froma vector ( σ ) and a square ( σ σ ), an oriented square ( σ σ ) from two vectors ( σ and σ ), a square ( σ σ ) from acube ( σ σ σ ) and a vector ( σ ), and so on. Furthermore, entangled quantum states are replaced by multivectorswith a clear geometric interpretation. For instance, a general multivector M in Cl (3) is a linear combination of blades,geometric products of different basis vectors supplemented by the identity 1 (basic oriented scalar), M def = M X j =1 M j σ j + X j C. C. thanks C. Lupo and L. Memarzadeh for very useful discussions. 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