$Id: QuCalc.help,v 1.1 2000/10/02 18:49:26 glenn Exp$ Welcome to QuCalc, the quantum computation package for Mathematica 4.0. Author: Paul Dumais, dumais@iro.umontreal.ca Help messages: Paul Dumais and Hugo Touchette "Mathematica" is a registered trademark of "Wolfram Research". QuCalc may and must be distributed freely. It must be distributed without modifications, including the name of the authors and this message. Type "?intro" or "?list" for help. General remarks relative to QuCalc ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ QuCalc is a Mathematica package whose pupose is to simulate and solve problems related to quantum computing. QuCalc has been developed by Paul Dumais at Laboratoire d'informatique thŽorique et quantique (MontrŽal University) and at the Crypto and Quantum Info Lab (McGill University). A minimal knowledge of Mathematica is required in order to use the QuCalc package effectively. It is useful to recall that Mathematica does not incorporate in its kernel a data type for matrices: instead, one must use a "list of lists" to represent matrices. It is strongly recommended to carry out and analyze the examples suggested in the help sections of the this package. These examples form an essential complement to the main help text. The paradigm adopted for the QuCalc Mathematica package is that of converting states and operators into vectors and matrices. Kets are represented as column vectors written in the canonical basis (i.e. the basis into which the Pauli operator \sigma_z is diagonal); unitary transformations are represented as square matrices; mixed states correspond to density matrices; etc. The identifiers defined when QuCalc is loaded in Mathematica all begin with lowercase letters. Following standard usage, identifiers beginning with a capital letter are those of Mathematica. Some identifiers of Mathematica are also overloaded in QuCalc such as "CircleTimes". Whenever you don't know how to interpret a result x returned by QuCalc, try FullForm[x]. This will tell you how QuCalc "interprets" the result. If x appears more complicated than it should be, test on it several simplification tools included in Mathematica, such as Simplify[x], FullSimplify[x], TrigReduce[x], ComplexExpand[x], ... In general, functions do not test their inputs. When invalid inputs are supplied to a function, the result is not defined (error message from Mathematica, bizarre answer, etc.). To obtain help pertaining to a specific function f, type "?f". To obtain the list of the subjects for which help is supplied, type "?list". Errors or bugs found in the QuCalc package, as well as questions or comments should be addressed to Paul Dumais, dumais@iro.umontreal.ca. General information: intro list Mathematica identifiers overloaded in QuCalc: CircleDot CirclePlus CircleTimes Dot OverBar OverTilde Power SuperDagger Vee Wedge Data types: ens ket schmidt sqo state supop unit Tests: ensQ ketQ sqoQ stateQ supopQ unitQ Constants: bb84 cnot knot mm mmm not phim phip psim psip sigx sigy sigz wh xm xp ym yp zm zp Functions: anc band bits block bnot bor bscal bxor circuit cycle ctrl dag dotexp eigen eigenVal eigenVect entropy fgate fidelity fourier gate id kron krondiv kronexp ktrl maxmix phase randomUnit rotx roty rotz swap trout unvec vec Dot[x, y, ...] x.y. ... Dot gives the product (composition, application) of its arguments. The arguments may be in the form of matrices, "ket", "unit", "state", etc, as long as the product is defined. Note that if u is a "unit" and r is a "state", then u.r returns the density matrix resulting from the application of u on r. Do not type u.r.dag[u]. Try: dag[xp] . zp sigz . zm sigz . sigz wh . state[zm] mmm . xp wh . mm . wh See also: circuit, dotexp, kron OverBar[x] Returns the complex conjugate of x. It is equivalent to the Mathematica function "Conjugate". This function can be entered using 2D notation by typing: "CTRL-&" "_". Try: OverBar[I] See also: dag anc[nA, nB, nC] Returns a data structure of type "sqo" representing the operation of adding a constant ancillary state |0> of dimension nB in between two Hilbert spaces of dimensions nA and nC. Try: r = anc[2,2,2] . ket["11"] eigen[r] See also: sqo, trout band[s1, s2, ...] Wedge[s1, s2, ...] Performs a bitwise "and" operation over binary strings, i.e., lists of 0's and 1's. This function may be entered in infix notation using the operator "Wedge" of Mathematica, by typing: "ESC" "^" "ESC". Try: band[{0,1,1,1}, {0,0,1,1}] See also: bits, bnot, bor, bscal, bxor bb84 Constant of type "ens" representing a statistical ensemble of 4 equiprobable states corresponding to the qubits 0 and 1 written in the canonical and diagonal bases. Try: bb84 wh . bb84 state[bb84] See also: ens, zp bits[x, n] Converts the integer x into its corresponding binary expression (list of 0's and 1's). Try: ket[bits[5,3]] ket[bits[5,4]] See also: band, bnot, bor, bscal, bxor block[u, v] Block product of the unitary transformations u and v. Try: block[wh, wh] See also: kron, unit bnot[s] OverTilde[s] Returns the binary complement (negation) of a bit list. This function can be entered in 2D notation by typing: "CTRL-&" "~". Try: bnot[{0,1,0}] See also: band, bits, bor, bscal, bxor bor[s1, s2, ...] Vee[s1, s2, ...] Performs a bitwise "or" operation over bit lists. This functions can be called in "infix" notation using the Mathematica operator "Vee" by typing: "ESC" "v" "ESC". Try: bor[{0,1,1,1}, {0,0,1,1}] See also: band, bits, bnot, bscal, bxor bscal[s1, s2] CircleDot[s1, s2] Returns the boolean scalar product of the binary lists s1 and s2. This function can be called using the Mathematica infix operator "CircleDot", by typing: "ESC" "c" "." "ESC". Try: bscal[{0,1,1,1}, {0,0,1,1}] See also: band, bits, bnot, bor, bxor bxor[s1, s2, ...] CirclePlus[s1, s2, ...] Returns the result of a bitwise "exclusive or" operation performed over bit lists. This function may be called in infix notation using the Mathematica operator "CirclePlus" by typing: "ESC" "c" "+" "ESC". Try: bxor[{0,1,1,1}, {0,0,1,1}] See also: band, bits, bnot, bor, bscal circuit[m] This function performs a series of compositions and Kronecker products associated with the elements of the matrix m. The lines of m represent the quantum wires which carry qubits from left to right. The columns of m represent the quantum gates to be applied on qubits. In other words, the matrix m is a gate array. For example, circuit[{{wh,id[]}, {not,not}}] returns a data structure of type "unit" (a unitary transformation) equivalent to a circuit of two gates acting on two quantum wires. The first wire is tranformed by a Walsh-Hadamard (wh) operation followed by a negation (not) operation. The second wire, is left unchanged by the first identity (id[]) operation, and is then negated by a "not" operation. To build a gate acting coherently on a subset of selected qubits (leaving unchanged the rest of the qubits), place the desired transformation on the last qubit of the selected subset in question. The other wires are assigned integers describing a "chained list" in top-down order. For example, circuit[{{id[]},{1},{3},{id[]},{id[]},{ctrl[ctrl[not]]}}] represents a Toffoli gate (ctrl[ctrl[not]]), acting on the 2nd, 3rd and 6th wires of a quantum circuit of 6 qubits. The function "circuit" is very useful when combined with the 2D notation of Mathematica commonly used to create matrices: "CRTL-RETURN" to create the lines, and "CTRL-," to create the columns. In that case, the circuit entered in Mathematica looks exactly as in standard gate array notation. Note also that the "Placeholder" of Mathematica (the little blank square appearing in 2D notation), stands for the identity transformation. It is thus unnecessary to fill out all the entries of a circuit. Try: circuit[{{ 1, 1, 1 }, {cnot, knot, cnot}}] circuit[{{ 1, mm}, {cnot, mm}}] See also: Dot, ctrl, cycle, fgate, gate, kron, ktrl, swap phim phip psim psip xm xp ym yp zm zp Constants of type "ket". phim, phip, psim, psip: Bell's states. xm, xp, ym, yp, zm, zp: Pure states corresponding to the poles and the 4 cardinal points on the equator of the Bloch-PoincarŽ sphere. Try: cnot . kron[wh, id[]] . ket["00"] == phip See also: ket cnot knot not sigx sigy sigz wh Constants of type "unit". cnot: controled-not; knot: inversed controled-not; not: negation (of 1 qubit); sigx, sigy, sigz: Pauli matrices; wh: Walsh-Hadamard transformation. Try: not == sigx See also: unit ctrl[u] ctrl[n, u] ktrl[u] Various forms of controlled gates (conditional operations). ctrl[u]: controlled u gate; ctrl[n,u]: u gate controlled by n wires; ktrl[u]: inverted controlled u gate. Try: ctrl[not] == cnot ctrl[2,not] ktrl[wh] See also: circuit, cnot, knot cycle[n,i,j] Unitary transformation acting on n qubits which performs a circular permutation of the qubits i to j (in top-down order). Try: c = cycle[3,1,3] c . ket["010"] == ket["001"] Power[c,2] == dag[c] See also: circuit dag[x] SuperDagger[x] Returns the conjugate transpose of x. This function can be called in 2D notation by typing: "CTRL-^" "ESC" "d" "g" "ESC". Try: dag[wh] == wh dotexp[x, n] Power[x, n] x^n These operators return the result of the n-fold product x.x. ... .x. The argument x can be of type "unit", "supop", "sqo", etc., as long as the result is meaningful. Try: Simplify[rotx[t]^3] See also: Dot, kronexp eigen[r] eigenVal[r] eigenVect[r] Returns the eigenvalues and eigenvectors of a state. eigen[r]: returns a statistical ensemble (type "ens") formed with all the pairs eigenvalues-eigenvectors of r. eigenVal[r]: returns a diagonal matrix whose elements are the eigenvalues of the state r, in the data type "state". eigenVect[r]: returns a unitary matrix whose columns are the eigevectors of r. Try: eigen[maxmix[2]] eigenVect[state[xp]] eigenVal[state[xp]] eigen[state[yp]] == yp eigen[bb84] eigen[state[bb84]] See also: ens, schmidt, state, unit ens[...] Data type representing a statistical ensemble of states. A valid "ens" object contains a list of pairs {p,r} where p is a real number between 0 and 1 (a probability), and r is an object of type "ket" or "state". The p's must add to unity. An "ens" object containing a trivial list of only one pair {p,r} with p=1 is automatically converted to a "state" or "ket" r. Try: bb84 FullForm[bb84] mmm . xp FullForm[mmm . xp] eigen[maxmix[2]] FullForm[eigen[maxmix[2]]] See also: bb84, eigen, ensQ, ket, state, sqo entropy[r] Numerical calculation of the von Neumann entropy of the state r. Try: entropy[state[bb84]] See also: fidelity fgate[m, n, f] Returns a unitary transformation, acting on m+n qubits, that is equivalent to the function f acting on classical inputs. The function f is a Mathematica "pure function". It takes on input a list of m bits and must return a list of n bits. Thus, f is a (classical) function of m bits to n bits. Try: fgate[1,1,(#)&] fgate[1,2,({#[[1]], #[[1]]})&] See also: circuit, gate fidelity[r1, r2] Numerical calculation of the fidelity function of two (pure or mixed) states. Try: fidelity[zp, xp] See also: entropy fourier[m] m-dimensional Fourier transform. Try: fourier[4] See also: circuit gate[n, b, f] gate[n, f] Returns a unitary transformation of dimension b^n (2^n if b is ommitted) described by the function f. The function f is a Mathematica "pure function". It takes as input n integers between 0 and b-1, and returns a "ket". For different values in its arguments f must return orthogonal kets. Try: gate[2,(ket[{#1, bxor[#1,#2]}])&] See also: circuit, fgate id[n] id[] Identity transformation on n qubits (1 if n is ommited). Try: id[8] See also: circuit ket[...] Data type representing a pure state. A well-defined "ket" contains a matrix of dimension n x 1, and must be of norm 1. Note that in 2D notation, lines of a matrix can be created using "CTRL-RETURN". To obtain a "bra" from a "ket", use dag[ket[...]]. There is no data type "bra" in QuCalc. The expression ket[0] (or ket[1]) is converted in such a way to represent the bit 0 (or 1) in the canonical basis of a 2-dimensional Hilbert space, i.e., a qubit. To obtain a register of many qubits, one can concatenate several bits into a list or into a character string. To obtain a quantum register whose dimension is a power not reducible in base 2, one can add as a subscript the value of the base in question. See the examples below. In 2D notation, "Subscript[x,y]" may be entered by typing: "x" "CTRL-_" "y". Try: ket[0] FullForm[ket[0]] FullForm[ket[0][[1]]] ket[{0,1,0}] ket["010"] ket[Subscript[2, 3]] ket[Subscript[{0,2,0}, 3]] ket[Subscript["020", 3]] -zm ket[(1/Sqrt[2])(zp+zm)] See also: ketQ, phip, zp kron[x, y, ...] CircleTimes[x, y, ...] Returns the Kronecker product of x,y,... The arguments can be matrices, "ket", "unit", "state", etc., as long as the product is meaningful. This function in Mathematica can be called in infix notation using the operator "CircleTimes" by typing: "ESC" "c" "*" "ESC". Note that this operator has a weaker precedence than the "Dot" product, which denoted with ".". Try: kron[zp, xp] kron[wh, cnot] kron[zm, maxmix[2]] kron[mm, mm] kron[mm, wh] See also: Dot, circuit, krondiv, kronexp krondiv[v, w] "Kronecker division". krondiv returns a matrix y of dimension N/n x 1, such that v = kron[w,y], where v is a matrix of dimension N x 1, and w is matrix of dimension n x 1. The function krondiv only accepts Mathematica matrices, and no other types of data, such as "ket". Note that lines of a column vector can be created in Mathematica using the 2D notation: "CTRL-RETURN". Try: krondiv[{{4},{5},{8},{10},{12},{15}}, {{1},{2},{3}}] See also: kron, vec, unvec kronexp[x, n] n-fold Kronecker product. Try: kronexp[wh, 3] kronexp[mm, 3] kronexp[zp, 3] == ket["000"] See also: kron, dotexp maxmix[n] Returns the maximally mixed density matrix of dimension n. Try: maxmix[3] See also: state mm Constant of type "supop" representing a 1-qubit measuring process in the canonical basis. Try: mm mm . xp mm . zp kron[mm, mm] mm . mm == mm wh . mm . wh See also: mmm, supop mmm Constant of type "sqo" representing a 1-qubit measuring process in the canonical basis. Try: mmm mmm . xp mmm . zp kron[mmm, id[]] mmm . mmm == mm See also: mm, sqo randomUnit[] Returns a random unitary transformation of dimension 2. The distribution of the outcome is based on a uniform random choice of the Euler angles in the Bloch-PoincarŽ sphere. More precisely, randomUnit[] is defined as follows: phase[t1] . rotz[t2] . roty[t3] . rotz[t4] where t1, t2, t3, and t4 are independent and identically distributed uniform random variables chosen between 0 and 2 Pi. Try: randomUnit[] . zp See also: phase, unit phase[t] rotx[t] roty[t] rotz[t] phase[t] is a global phase change with angle t applied to 1 qubit. rotx[t], roty[t] and rotz[t] are rotations around the axis corresponding to the function name in the Bloch-PoincarŽ sphere. Try: phase[t] rotx[t] roty[t] rotz[t] See also: unit, randomUnit schmidt[...] Data type representing a Schmidt decomposition of a pure state seen as a bipartite state. A well-defined "schmidt" contains a list of triplets {q, k1, k2}, where the q's are the Schmidt coefficients of the decomposition, the sum of the q^2 must be 1. The k1's (the k2's) are objects of type "ket", which form an orthonormal basis, the Schmidt basis, of Alice's (Bob's) sub-space. The expression schmidt[n1,n2,k], where k is a "ket" of dimension n1*n2 is converted so as to represent a Schmidt decomposition of k with n1 and n2 as the respective dimensions of the spaces of the two parties (Alice and Bob). A fourth optional argument, of Mathematica type "pure function", may be passed. It is a simplification function that will be called at certain stages of the computation. The call schmidt[n1, n2, k] is equivalent to schmidt[n1, n2, k, (Simplify[#])&]. Try: s = schmidt[2,2,phip] FullForm[s] FullForm[s[[1]]] ket[s] See also: ket, eigen sqo[...] Data type representing a "selective quantum operation" (SQO). Mathematically, a SQO is defined as a family of square or rectangular matrices A_{i,j}, 1<=i<=m, 1<=j<=k_i, of dimension s x r, such that \sum A_{i,j}^\dagger A_{i,j} is equal to the r x r identity matrix. A SQO is a general quantum operation which includes, as special cases, unitary transformations, superoperators, POVMs (Positive Operator Valued Measure), trace-out's, or the effect of adding ancillae. When applied to a mixed state \rho, of arbitrary dimension r, a SQO returns with probability p_i the state \sigma_i = (1/p_i) B_i of dimension s, where B_i = \sum_{j} A_{i,j} \rho A_{i,j}^\dagger p_i = tr(B_i). It is very unlikely that you will have to build a "sqo" entirely from scratch, except possibly for the case where a measurement along non-orthogonal axes has to be made. In general, it is sufficient to combine the sqo's already defined in QuCalc with some unitary transformations to obtain a general quantum operation. To get acquainted with the details of a "sqo", refer to the examples given below. Try: mmm FullForm[mmm] q = kron[mmm, mm] FullForm[q] mmm . xp See also: anc, mmm, sqoQ, supop, trout state[...] Data type representing (in general) a mixed state. A well-defined "state" must contain a positive hermitian square matrix with trace equal to one. Note that in 2D notation, "CTRL-RETURN" can be used to create the lines of a matrix, whereas "CTRL-," is used to create the columns. An object "state" representing a pure state is not automatically converted into a "ket"; use "eigen" to force that conversion. Try: r = state[phip] FullForm[r] FullForm[r[[1]]] state[(1/2)(state[phip] + state[phim])] See also: eigen, ket, maxmix, stateQ supop[...] Data type representing a superoperator. A well-defined "supop" must contain a list of square matrices A_j such that \sum_j A_j^\dagger A_j is equal to the identity matrix. When applied to a mixed state \rho, a "supop" returns the mixed state \sum_{j} A_j \rho A_j^\dagger as long as the dimensions of the matrices are compatible. In 2D notation, "CTRL-RETURN" can be used to create the lines of a matrix, whereas "CTRL-," is used to create the columns. Try: mm FullForm[mm] mm . xp See also: mm, sqo, supopQ swap[n, i, j] swap[] Returns a unitary transformation acting on n qubits, and whose action is to swap qubits i and j. swap[] is equivalent to swap[2,1,2]. Try: c = swap[4,1,3] c . ket["0110"] == ket["1100"] Power[c,2] == id[4] See also: circuit ensQ[x] ketQ[x] sqoQ[x] stateQ[x] supopQ[x] unitQ[x] Boolean functions testing the validity of x according to the type structure associated with the function name. Try: unitQ[wh] Simplify[unitQ[fourier[3]]] ketQ[ket[zp + zm]] See also: ens, ket, sqo, state, supop, unit trout[nA, nB, nC] This functions returns a "sqo" data structure representing the operation of tracing out a Hilbert space of dimension nB in between 2 Hilbert spaces of dimension nA and nC. Try: trout[2, 2, 1] . phip trout[1, 2, 2] . phip trout[2, 4, 2] . ket["0110"] == state[ket["00"]] See also: sqo, anc unit[...] Data type corresponding to unitary transformations. A valid "unit" data is a square unitary matrix. Note that a matrix in 2D notation can be built using "CRTL-RETURN" to create lines and "CTRL-," to create columns. Try: FullForm[sigz] FullForm[sigz[[1]]] I wh unit[Exp[(I Pi)/4] wh] unit[(1/Sqrt[3])(sigx + sigy + sigz)] See also: cnot, fourier, id, knot, not, phase, randomUnit, rotx, sigx, unitQ, wh vec[x] unvec[v, n] This function transforms a matrix x into a vector. The columns of the matrix (from left to right) are concatenated in the vector in top-down order. unvec[v, n] undo the transformation by re-transforming a vector v into a matrix containing n lines. These functions are defined for Mathematica matrices; they cannot be applied to a "ket" or any other data types specific to QuCalc. In 2D notation "CTRL-RETURN" can be used to create lines of a matrix, whereas "CTRL-," is used to create the columns. Try: v = vec[{{1,2,3},{4,5,6},{7,8,9},{10,11,12}}] FullForm[v] unvec[v, 2] See also: krondiv