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Lecture Notes: Qubit Representations and Rotations

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Lecture Notes: Qubit Representations and Rotations Phys 711 Topics in Particles & Fields | Spring 2013 | Lecture 1 | v0.3 Lecture notes: Qubit representations and rotations Jeffrey Yepez Department of Physics and Astronomy University of Hawai`i at Manoa Watanabe Hall, 2505 Correa Road Honolulu, Hawai`i 96822 E-mail: [email protected] www.phys.hawaii.edu/∼yepez (Dated: January 9, 2013) Contents mathematical object (an abstraction of a two-state quan- tum object) with a \one" state and a \zero" state: I. What is a qubit? 1 1 0 II. Time-dependent qubits states 2 jqi = αj0i + βj1i = α + β ; (1) 0 1 III. Qubit representations 2 A. Hilbert space representation 2 where α and β are complex numbers. These complex B. SU(2) and O(3) representations 2 numbers are called amplitudes. The basis states are or- IV. Rotation by similarity transformation 3 thonormal V. Rotation transformation in exponential form 5 h0j0i = h1j1i = 1 (2a) VI. Composition of qubit rotations 7 h0j1i = h1j0i = 0: (2b) A. Special case of equal angles 7 In general, the qubit jqi in (1) is said to be in a superpo- VII. Example composite rotation 7 sition state of the two logical basis states j0i and j1i. If References 9 α and β are complex, it would seem that a qubit should have four free real-valued parameters (two magnitudes and two phases): I. WHAT IS A QUBIT? iθ0 α φ0 e jqi = = iθ1 : (3) Let us begin by introducing some notation: β φ1 e 1 state (called \minus" on the Bloch sphere) Yet, for a qubit to contain only one classical bit of infor- 0 mation, the qubit need only be unimodular (normalized j1i = the alternate symbol is |−i 1 to unity) α∗α + β∗β = 1: (4) 0 state (called \plus" on the Bloch sphere) 1 Hence it lives on the complex unit circle, depicted on the j0i = the alternate symbol is j+i: 0 top of Figure 1. Normalization (4) constrains the value of the magnitudes, so we can write a qubit as When you learn about the Bloch sphere (discussed below) p 1 − f you will see why the alternate symbols j+i and |−i are jqi = p ; (5) f ei' used to denote logical states.1 So what is a qubit? A qubit is the fundamental quan- where 0 ≤ f ≤ 1 and where an irrelevant overall phase is tum state representing the smallest unit of quantum in- factored out. The length (or norm) of the qubit is thus formation containing one bit of classical information ac- an invariant quantity cessible by measurement. We simply take a qubit to be a hqjqi = jαj2 + jβj2 = jp1 − fj2 + jpfj2 = 1: (6) You should understand why an overall phase is irrelevant 1 The names \up" and \down," and the respective symbols j"i 1 to the length of the qubit. The quantum property of and j#i, are reserved for spin- 2 particles.We will see in another lecture how a 2-qubit encoding conforms with the Pauli exclusion measurement follows from identifying the moduli squared principle for particles with half-integer spin. of the amplitude as an occupation probability f and 1 − 2 f for the qubit to occupy its logical states j1i and j0i, Writing our 2-spinor basis states in terms of qubit states, respectively, as follows: we have θ 2 i ' cos f = jβj (7) ξ(") ≡ e 2 j+i = 2 ei' sin θ 1 − f = jαj2: (8) 2 θ θ = cos j0i + sin ei'j1i; (14a) There are only two relevant free parameters to specify the 2 2 ' −i' θ state of a qubit, but upon measurement, the qubit orig- −i −e sin 2 ξ(#) ≡ e 2 |−i = θ inally in the superposition state (5) is found to occupy cos 2 only one of its logical states θ θ = − sin e−i'j0i + cos j1i: (14b) ( 2 2 j1i; with probability f; jqi −−−−−!measure (9) j0i; with probability 1 − f: III. QUBIT REPRESENTATIONS Thus, upon a single measurement, jqi is found to be in either the state j0i or j1i, an outcome that is said to + = 0 È1\ ` be specified by a single classical bit 2 f0; 1g. Thus in z actual experiments, the occupation probability f equals J Èq\ ãä j sin » \ » \ the frequency of occurrence of the result 1 obtained from 2 J q many repeated measurements. 1 J È0\ » \ J ` cos y 2 j II. TIME-DEPENDENT QUBITS STATES ` x The state jq(t)i of a time-dependent qubit, as a two- ` energy level quantum mechanical entity, is governed by -z the Schroedinger wave equation - = 1 FIG. 1 A qubit in Hilbert space in its SU(2) representation @ ~! » \ » \ i~ jq(t)i = σzjq(t)i: (10) (top), and the same qubit on the Bloch sphere in its O(3) @t 2 representation (bottom). SU(2) and O(3) are homomorphic. The energy eigenvalues are ±~!=2 and energy eigenstates are 1 0 A. Hilbert space representation j0i ≡ j1i ≡ ; (11) 0 1 The space of all possible orientations of jqi on the com- where j0i is the ground state and j1i is the excited state plex unit circle is called the Hilbert space. In the logical of the qubit. In terms of the angular frequency ! (e.g. basis, the two degrees of freedom of the qubit is often 2 θ Rabi frequency), the time-dependent qubit state is expressed as two angles θ and ', where f = sin 2 . So without any loss of generality the Hilbert space represen- −i ! t i ! t tation of a qubit (1) can be written as jq(t)i = A0e 2 j0i + A1e 2 j1i; (12) 2 θ θ i' where the complex probability amplitudes satisfy jA0j + jqi = cos j0i + sin e j1i: (15) 2 2 2 jA1j = 1 since the qubit resides on the complex circle in Hilbert space (or the Bloch sphere in spin space). These angles have a well known geometrical interpreta- Now, we can explicitly write out qubit basis states of tion as Euler angles. the Bloch sphere withu ^ = (sin θ cos '; sin θ sin '; cos θ) as B. SU(2) and O(3) representations θ −i ' 2 θ ' θ ' cos 2 e −i i j+iu = ' = cos e 2 j0i + sin e 2 j1i; θ i 2 sin 2 e 2 2 To understand the geometrical interpretation of a (13a) qubit, consider a three-dimensional space with \unit vec- θ −i ' tors" σx, σy, and σz chosen as an orthonormal basis. In 2 θ ' θ ' − sin 2 e −i i |−iu = ' = − sin e 2 j0i + cos e 2 j1i: quantum information theory, one represents each basis θ i 2 cos 2 e 2 2 element by a 2 × 2 matrix, a traceless hermitian genera- (13b) tors of two-dimensional special unitary group, SU(2). To 3 do so, one defines the symmetric product (dot product) In this representation, the qubit is expressed as a matrix as element of the SU(2) group. In quantum information, 1 usually 2×2 unitary matrices are considered single-qubit σi · σj ≡ σi · σj + σj · σi : (16a) quantum gates, but such matrices can themselves repre- 2 sent qubits too. Table I gives a summary of the three Furthermore, one defines the anti-symmetric product qubit representations (cross product) as Representations Qubit i Hilbert space jqi = cos ` θ ´ j0i + sin ` θ ´ ei'j1i σ × σ ≡ − σ · σ − σ · σ : (16b) 2 2 i j 2 i j j i O(3) group ~q = (sin θ cos '; sin θ sin '; cos θ) „ cos θ e−i' sin θ« SU(2) group M = Note that the centered dot symbol on the R.H.S. of (16) q ei' sin θ − cos θ denotes matrix multiplication. Thus, a basis that is or- thonormal satisfies the following conditions TABLE I Qubit representations. ( 1; for i = j (normal); σ · σ = (17a) i j 0; otherwise (orthogonal); IV. ROTATION BY SIMILARITY TRANSFORMATION and ( Now that we see a qubit as simply a unit vector on 0; for i = j; the complex circle (Hilbert space representation) or a σi × σj = (17b) σk; for cyclic indices: unit vector on the Bloch sphere (O(3) representation), we can consider rotations of the qubit's state that keep A fundamental matrix representation that satisfies (17) its length (or norm) invariant. Remarkably, such a rota- is the well-known Pauli basis tion of a qubit is conveniently accomplished by employ- 0 1 0 −i 1 0 ing its SU(2) representation as a 2 × 2 unitary matrix. σ = ; σ = ; σ = : 1 1 0 2 i 0 3 0 −1 Then, the qubit rotation is induced by a similarity trans- (18) formation, which is to say a double-sided transformation Exercise: acting from the left and the right side. The unitary ma- trix (acting from the left) along with its matrix inverse If it is not obvious already, verify that the (acting from the right) that is customarily employed for Pauli matrices (18) satisfy the orthonormal- such rotations, about the ith principle axis say, is ity conditions (17) which is just the structure −i θ σ θ θ equation for the SU(2) group U (θ) ≡ e 2 i = σ cos − iσ sin ; (22) i 0 2 i 2 [S ;S ] = i S ; i j ijk k 1 0 σi where the identity matrix is σ0 = . Explicitly, the where Si ≡ 2 and the structure constant ijk 0 1 is the anti-symmetric Levi-Civita symbol. unitary matrices for the principles directions are cos θ −i sin θ U (θ) = 2 2 (23a) 1 −i sin θ cos θ Now we can express the qubit (15) in vector form (i.e.
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