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<html>
<head>
<title>Muqcs</title>
<meta charset="utf-8"/>
</head>
<body>
Open a console (in Chrome, this is done by selecting 'Developer Tools'). From the console prompt, you can call functions in the code and see output printed to the console.
<p>
Or go to the <a href="https://github.com/MJMcGuffin/muqcs">github repository</a> page.
</p>
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjs/13.0.2/math.js"></script>
<script>
// CONVENTIONs
//
// Assume that, in a quantum circuit with n qubits, the qubits are numbered
// from top to bottom as 0 to (n-1), respectively, and furthermore that the
// top qubit is the least significant bit (LSB), and the bottom is the most
// significant bit (MSB). For example, if there are 3 qubits labeled q0, q1, q2,
// from top to bottom, these might store (q2,q1,q0) = (0,1,1) to encode 3
// or (q2,q1,q0) = (1,0,0) to encode 4. Given these assumptions, there are
// two ways to simulate the circuit, described below, that correspond to
// different matrices and different orderings for tensor products.
// Sometimes the terms "big-endian" or "little-endian" are used to refer
// to choices related to this, but I haven't found clear definitions
// or consistent usage of these terms.
//
// Common software convention:
// - Tensor products are done in order of decreasing bit significance.
// https://quantumcomputing.stackexchange.com/questions/8244/big-endian-vs-little-endian-in-qiskit
// seems to confirm that Qiskit works this way.
// - In a 2-qubit circuit, if we have a 2-qubit gate where q0 is a control
// bit and q1 has a conditionally applied operation with 2x2 matrix [a b; c d],
// the 4x4 matrix for the gate would be [1 0 0 0; 0 a 0 b; 0 0 1 0; 0 c 0 d],
// which is endian-reversed with how 4x4 matrices are usually presented in
// textbooks on quantum computing.
// See https://qiskit.org/documentation/tutorials/circuits/3_summary_of_quantum_operations.html#Basis-vector-ordering-in-Qiskit
// - In a 3-qubit circuit, the state vector would contain 8 elements,
// and the 2nd of these would correspond to (q2,q1,q0) = (0,0,1),
// which is more natural for reading. This appears to match the
// presentation of output from Quirk and IBM Quantum Composer.
// Quirk's doc/README.md file ( https://github.com/Strilanc/Quirk/blob/master/doc/README.md )
// states "Kets are big-endian. |00101> is 5, not 20",
// and https://algassert.com/post/1707 further discusses this.
//
// Textbook convention:
// - Tensor products are done in order of increasing bit significance.
// This seems to match when
// https://qiskit.org/documentation/tutorials/circuits/3_summary_of_quantum_operations.html#Basis-vector-ordering-in-Qiskit
// states "Within the physics community, the qubits of a multi-qubit systems
// are typically ordered with the first qubit on the left-most side of the
// tensor product and the last qubit on the right-most side."
// - The same 4x4 matrix mentioned earlier, with q0 acting as a control bit,
// would now be [1 0 0 0; 0 1 0 0; 0 0 a b; 0 0 c d] which is how they
// are usually presented in textbooks on quantum computing.
// - In a 3-qubit circuit, the state vector would contain 8 elements, and
// the 2nd of these would correspond to (q0,q1,q2) = (0,0,1).
// Such a vector would have to be endian-reversed for more natural reading.
//
// We could just implement one of the above conventions in the code,
// but I found it instructive to try to implement both and have a global flag
// to switch between the two.
//
let usingTextbookConvention = false;
let defaultDecimalPrecision = 3;
let precisionForApproximateComparison = 0.01;
class Util {
static assert( condition, message ) {
if ( ! condition ) {
console.log( "ASSERTION ERROR: " + message );
console.trace(); // causes line numbers to be printed
}
}
// Let n be a power of 2. This returns the reverse of i in binary, with respect to n.
// For example, if n===16, the reverse of 1 is 8, the reverse of 2 is 4, the reverse of 3 is 12,
// and the reverse of 0, 6, 9, and 15 leaves each of those numbers unchanged.
static reverseEndianness(i,n) {
Util.assert( 0<=i && i<n && 1<=n, `Util.reverseEndianness(): unexpected condition, i===${i}, n===${n}`);
let result = 0;
let bit = 1;
let reversed_bit = n >> 1;
while ( bit < n ) {
if ( i & bit )
result |= reversed_bit;
bit <<= 1;
reversed_bit >>= 1;
}
Util.assert( bit === n/*this should happen because n should be a power of 2*/, `Util.reverseEndianness(): unexpected condition, n===${n}`);
return result;
}
// Returns angle in [0,2 pi] of the given point in the cartesian plane measured counterclockwise+ with respect to x+
static angleIn2D(x,y) {
let hypotenuse = Math.sqrt(x*x+y*y);
let angle = 0;
if ( hypotenuse > 0 ) {
let sine = y / hypotenuse;
if ( sine>= 1 ) angle = Math.PI/2;
else if ( sine<=-1) angle = -Math.PI/2;
else angle = Math.asin(sine);
if ( x < 0 ) angle = Math.PI - angle;
if ( angle < 0 ) angle += 2*Math.PI; // ensures angle is in [0,2*M_PI]
}
return angle;
}
}
class StringUtil {
static numToString( x, decimalPrecision = defaultDecimalPrecision ) {
let s1 = x.toString();
let s2 = x.toFixed(decimalPrecision);
if ( s1.length < s2.length ) return s1;
else return s2;
}
// returns the given string, reversed
static reverseString(s) {
let r = "";
for ( let j = s.length-1; j >= 0; j-- )
r += s[j];
return r;
}
// returns s repeated n times
static repeatString(s,n) {
let r = "";
for ( let j = 0; j < n; j++ )
r += s;
return r;
}
// converts an integer to a binary string, padding with zeros at the front
static intToBinaryString( i, desiredWidth ) {
let binaryString = i.toString(2/*base 2 for binary*/);
return StringUtil.repeatString('0',desiredWidth-binaryString.length) + binaryString; // pad with '0's
}
// returns number of times the character c occurs in string s
static countInString(s,c) {
let r = 0;
for ( let j = s.length-1; j >= 0; j-- )
if ( s[j]===c )
r ++;
return r;
}
// Returns a single multiline string containing the given strings
// concatenated horizontally and centered vertically.
// Assumes that none of the given strings are jagged,
// i.e., assumes that for each given string, all its lines are the same length.
//
// For example, calling with arguments ("x = ", "[0,0,1,1]", " * ", "[0]\n[1]\n[0]\n[1]")
// causes a return of the string
// " [0]\n"
// +"x = [0,0,1,1] * [1]\n"
// +" [0]\n"
// +" [1]"
//
static concatMultiline(...args) {
let w = []; // widths of given strings
let h = []; // heights of given strings
let max_height = 0;
for (let s of args) {
let s2 = s.split('\n');
h.push( s2.length );
if ( s2.length > max_height ) max_height = s2.length;
w.push( s2[0].length );
}
let returnValue = "";
for ( let row = 0; row < max_height; ++row ) {
for (let stringIndex = 0; stringIndex < args.length; stringIndex ++ ) {
let s = args[ stringIndex ];
let rowWithinString = Math.round( row - max_height/2 + h[ stringIndex ]/2 );
if ( rowWithinString < 0 || rowWithinString >= h[ stringIndex ] ) {
// use spaces
returnValue += StringUtil.repeatString(' ', w[ stringIndex ] );
}
else {
//console.log("rowWithinString = " + rowWithinString + ", s =" + s + "END" );
returnValue += s.split('\n')[rowWithinString];
}
}
if ( row < max_height-1 )
returnValue += '\n';
}
return returnValue;
}
}
// Stores a complex number, with real and imaginary components.
class Complex {
constructor( re = 0, im = 0 ) {
this._r = re;
this._i = im;
}
toString( decimalPrecision = defaultDecimalPrecision ) {
if ( this._r === 0 ) {
if ( this._i === 0 ) {
return "0";
}
else {
return StringUtil.numToString(this._i,decimalPrecision) + "i";
}
}
else {
if ( this._i === 0 ) {
return StringUtil.numToString(this._r,decimalPrecision);
}
else {
let rs = StringUtil.numToString(this._r,decimalPrecision);
let is = StringUtil.numToString(this._i,decimalPrecision) + "i";
return is[0]==='-' ? ( rs+is ) : ( rs + "+" + is );
}
}
}
// returns a deep copy of the complex number
copy() {
return new Complex( this._r, this._i );
}
conjugate() {
return new Complex( this._r, - this._i );
}
mag() { // magnitude, also known as absolute value or modulus
return Math.sqrt( this._r * this._r + this._i * this._i );
}
magSquared() { // magnitude squared
return this._r * this._r + this._i * this._i;
}
arg() { // argument, also called phase, i.e. the angle of the complex number; always in [0,2 pi]
return Util.angleIn2D( this._r, this._i );
}
// Returns the sum of the two given complex numbers.
static sum(c1,c2) {
return new Complex( c1._r+c2._r, c1._i+c2._i );
}
// Returns the difference of the two given complex numbers.
static diff(c1,c2) {
return new Complex( c1._r-c2._r, c1._i-c2._i );
}
// Returns the product of the two given numbers.
static mult(c1,c2) {
if ( c1 instanceof Complex ) {
if ( c2 instanceof Complex ) {
return new Complex( c1._r*c2._r - c1._i*c2._i, c1._r*c2._i + c1._i*c2._r );
}
return new Complex( c1._r * c2, c1._i * c2 );
}
else if ( c2 instanceof Complex ) {
return new Complex( c1 * c2._r, c1 * c2._i );
}
return c1 * c2;
}
}
Complex.i = new Complex(0,1);
// This is a complex matrix, i.e., a matrix containing complex numbers.
// To save on overhead, the complex numbers are stored as consecutive pairs of numbers in a floating point number array.
class CMatrix {
// allocates space in memory containing zeros
allocate( numRows, numCols ) {
this._rows = numRows;
this._cols = numCols;
this._m = new Float32Array(
numRows * numCols
* 2 // because we're storing a real and imaginary component for each matrix element
);
}
constructor( numRows, numCols ) {
Util.assert( numRows > 0 && numCols > 0, "CMatrix.constructor(): invalid size requested" );
this.allocate( numRows, numCols );
}
// Gets the value in a cell.
// row and column indices are zero-based.
// returns a complex number.
get(row,col) {
let j = ( row * this._cols + col )*2;
return new Complex( this._m[j], this._m[j+1] );
}
// Sets the value in a cell.
// row and column indices are zero-based.
// The value passed in can be a (real) number or a complex number.
set(row,col,value) {
let j = ( row * this._cols + col )*2;
if ( typeof(value)==='number' ) {
this._m[j ] = value; // real component
this._m[j+1] = 0;
}
else if ( typeof(value)==='object' && value instanceof Complex ) {
this._m[j ] = value._r;
this._m[j+1] = value._i;
}
else Util.assert(false,"CMatrix.set(): unknown type");
}
// Returns a deep copy of the matrix
copy() {
let M = new CMatrix( this._rows, this._cols );
M._m.set( this._m ); // this copies the contents of one to the other
return M;
}
// Returns the transpose of the matrix
transpose() {
let M = new CMatrix( this._cols, this._rows );
for ( let r = 0; r < this._rows; ++r ) {
for ( let c = 0; c < this._cols; ++c ) {
M.set( c, r, this.get(r,c) );
}
}
return M;
}
conjugate() {
let M = new CMatrix( this._rows, this._cols );
for ( let r = 0; r < this._rows; ++r ) {
for ( let c = 0; c < this._cols; ++c ) {
M.set( r, c, this.get(r,c).conjugate() );
}
}
return M;
}
// Returns the transpose of the matrix, also known as adjoint matrix or transjugate
conjugateTranspose() {
let M = new CMatrix( this._cols, this._rows );
for ( let r = 0; r < this._rows; ++r ) {
for ( let c = 0; c < this._cols; ++c ) {
M.set( c, r, this.get(r,c).conjugate() );
}
}
return M;
}
// Returns a complex number equal to the trace (i.e., sum of diagonal elements) of a square matrix
trace() {
Util.assert( this._rows === this._cols, "CMatrix.trace(): matrix is not square" );
let returnValue = new Complex();
for ( let r = 0; r < this._rows; ++r ) {
returnValue = Complex.sum( returnValue, this.get(r,r) );
}
return returnValue;
}
reverseEndianness( actuallyReverseThings=true ) { // should only be called on a row vector, column vector, or matrix whose dimensions are powers of 2
if ( ! actuallyReverseThings )
return this; // never mind
let M = new CMatrix( this._rows, this._cols );
for ( let r = 0; r < this._rows; ++r ) {
let r2 = Util.reverseEndianness(r,this._rows);
for ( let c = 0; c < this._cols; ++c ) {
let c2 = Util.reverseEndianness(c,this._cols);
M.set( r2, c2, this.get(r,c) );
}
}
return M;
}
// Returns a multiline string, e.g.,
// a 4x2 matrix might yield "[0,1+2i]\n[1,1 ]\n[0,2i ]\n[1,0 ]"
toString(
// this syntax allows for something similar to named parameters
{ binaryPrefixes=false, decimalPrecision=defaultDecimalPrecision, suppressZeros=true/*makes large binary matrices easier to read*/, charToReplaceSuppressedZero='_' }={}
) {
let arrayOfPrefixStrings = [];
if ( binaryPrefixes ) {
let targetWidth = Math.ceil( Math.log2(this._rows) );
for ( let r = 0; r < this._rows; ++r ) {
let binaryString = StringUtil.intToBinaryString( r, targetWidth );
if ( usingTextbookConvention )
binaryString = '|' + StringUtil.reverseString( binaryString ) + '>';
else
binaryString = '|' + binaryString + '>';
arrayOfPrefixStrings.push( binaryString );
}
}
let arrayOfArraysOfStrings = [];
for ( let r = 0; r < this._rows; ++r ) {
arrayOfArraysOfStrings.push( [] );
for ( let c = 0; c < this._cols; ++c ) {
arrayOfArraysOfStrings[r].push( this.get(r,c).toString(decimalPrecision) );
}
}
let maxMaxWidth = 0;
for ( let c = 0; c < this._cols; ++c ) {
let maxWidth = 0;
for ( let r = 0; r < this._rows; ++r ) {
let w = arrayOfArraysOfStrings[r][c].length;
if ( w > maxWidth ) {
maxWidth = w;
if ( maxWidth > maxMaxWidth ) maxMaxWidth = maxWidth;
}
}
// now we know the max width for this column, so we can pad with spaces
for ( let r = 0; r < this._rows; ++r ) {
let w = arrayOfArraysOfStrings[r][c].length;
arrayOfArraysOfStrings[r][c] += StringUtil.repeatString(' ',maxWidth-w);
}
}
// now we have finished padding all the strings with spaces
// so we can build the multiline string
let returnValue = '';
for ( let r = 0; r < this._rows; ++r ) {
if ( binaryPrefixes )
returnValue += arrayOfPrefixStrings[r];
returnValue += '[';
for ( let c = 0; c < this._cols; ++c ) {
if ( suppressZeros && maxMaxWidth===1 && arrayOfArraysOfStrings[r][c]==='0' )
returnValue += charToReplaceSuppressedZero;
else
returnValue += arrayOfArraysOfStrings[r][c];
if ( c < this._cols-1 )
returnValue += ',';
}
returnValue += ']';
if ( r < this._rows-1 )
returnValue += '\n';
}
return returnValue;
}
// Returns the sum of the two given matrices.
static sum(a,b) {
Util.assert(a instanceof CMatrix && b instanceof CMatrix,"CMatrix.sum(): wrong type");
Util.assert(a._rows === b._rows && a._cols === b._cols, "CMatrix.sum(): incompatible dimensions" );
let M = a.copy();
for ( let j = M._m.length-1; j >= 0; j-- )
M._m[j] += b._m[j];
return M;
}
// Returns the difference of the two given matrices.
static diff(a,b) {
Util.assert(a instanceof CMatrix && b instanceof CMatrix,"CMatrix.diff(): wrong type");
Util.assert(a._rows === b._rows && a._cols === b._cols, "CMatrix.diff(): incompatible dimensions" );
let M = a.copy();
for ( let j = M._m.length-1; j >= 0; j-- )
M._m[j] -= b._m[j];
return M;
}
// Returns the product of the two given matrices,
// or of a matrix with a scalar.
static mult( a, b ) {
Util.assert(a instanceof CMatrix || b instanceof CMatrix,"CMatrix.mult(): wrong type");
if ( !( a instanceof CMatrix ) ) {
// swap a and b, so that a will be the matrix and b the scalar
let tmp = b;
b = a;
a = tmp;
}
if ( typeof(b)==='number' ) {
let M = a.copy();
for ( let j = M._m.length-1; j >= 0; j-- )
M._m[j] *= b;
return M;
}
else if ( b instanceof Complex ) {
let M = new CMatrix(a._rows,a._cols);
for ( let r = 0; r < M._rows; r++ ) {
for ( let c = 0; c < M._cols; c++ ) {
let product = Complex.mult(a.get(r,c),b);
M.set(r,c,product);
}
}
return M;
}
else if ( b instanceof CMatrix ) {
Util.assert(a._cols === b._rows, "CMatrix.mult(): matrices have incompatible dimensions" );
let M = new CMatrix(a._rows,b._cols);
for ( let r = 0; r < M._rows; r++ ) {
for ( let c = 0; c < M._cols; c++ ) {
let dotProduct = new Complex();
for ( let k = 0; k < a._cols; ++k ) {
// TODO this line creates 4 instances of Complex that ultimately aren't
// needed; this could be optimized by expanding and inlining the math ops.
dotProduct = Complex.sum(dotProduct,Complex.mult(a.get(r,k),b.get(k,c)));
}
M.set(r,c,dotProduct);
}
}
return M;
}
else Util.assert(false,"CMatrix.mult(): unknown type");
}
// The kronecker product, or tensor product, of two matrices.
static tensor(a,b,isReversed=false) {
Util.assert(a instanceof CMatrix && b instanceof CMatrix,"CMatrix.tensor(): wrong type");
if ( isReversed ) {
let tmp = a;
a = b;
b = tmp;
}
let numRows = a._rows * b._rows;
let numCols = a._cols * b._cols;
let M = new CMatrix(numRows,numCols);
for ( let ar = 0; ar < a._rows; ar++ ) {
for ( let br = 0; br < b._rows; br++ ) {
for ( let ac = 0; ac < a._cols; ac++ ) {
for ( let bc = 0; bc < b._cols; bc++ ) {
M.set(
ar*b._rows + br,
ac*b._cols + bc,
Complex.mult( a.get(ar,ac), b.get(br,bc) )
);
}
}
}
}
return M;
}
// Imagine you want to form the product of many matrices m1 x m2 x ... mN
// You can obtain this product by calling the below routine with argument [m1,m2,...,mN]
//
// Since matrix multiplication is associative, we have a choice of computing the product
// starting with whatever matrices we like.
// It turns out to be more efficient to compute the product of smaller matrices first.
// So, this routine searches through the given list for the consecutive pair
// of smallest matrices, replaces them with their product, and repeats.
static naryMult( list ) {
Util.assert( list.length>0 && list[0] instanceof CMatrix, "CMatrix.naryMult(): invalid input" );
// let totalCost = 0;
while ( list.length > 1 ) {
let lowestCost = 0;
let indexForLowestCost = -1;
for ( let i = 0; i < list.length-1; ++i ) {
// compute the cost of computing the product of the ith and (i+1)th matrices
let cost = list[i]._rows * list[i]._cols * list[i+1]._cols;
if ( indexForLowestCost < 0 || cost < lowestCost ) {
lowestCost = cost;
indexForLowestCost = i;
}
}
// Replace the lowest-cost pair of matrices with a single matrix
let a = list[ indexForLowestCost ];
let b = list[ indexForLowestCost + 1 ];
// replace the ith and (i+1)th matrices with their product
list.splice( indexForLowestCost, 2, CMatrix.mult(a,b) );
// totalCost += lowestCost;
}
// console.log("totalCost: " + totalCost);
return list[0];
}
// Imagine you want to form the tensor product of many matrices m1 x m2 x ... mN
// You can obtain this product by calling the below routine with argument [m1,m2,...,mN]
//
// Since the tensor product is associative, we have a choice of computing the product
// starting with whatever matrices we like.
// It turns out to be more efficient to compute the product of smaller matrices first.
// So, this routine searches through the given list for the consecutive pair
// of smallest matrices, replaces them with their product, and repeats.
static naryTensor( list, isReversed=false ) {
Util.assert( list.length>0 && list[0] instanceof CMatrix, "CMatrix.naryTensor(): invalid input" );
if ( isReversed ) {
list.reverse();
}
// let totalCost = 0;
while ( list.length > 1 ) {
let lowestCost = 0;
let indexForLowestCost = -1;
for ( let i = 0; i < list.length-1; ++i ) {
// compute the cost of computing the product of the ith and (i+1)th matrices
let cost = list[i]._rows * list[i]._cols * list[i+1]._rows * list[i+1]._cols;
if ( indexForLowestCost < 0 || cost < lowestCost ) {
lowestCost = cost;
indexForLowestCost = i;
}
}
// Replace the lowest-cost pair of matrices with a single matrix
let a = list[ indexForLowestCost ];
let b = list[ indexForLowestCost + 1 ];
// replace the ith and (i+1)th matrices with their product
list.splice( indexForLowestCost, 2, CMatrix.tensor(a,b) );
// totalCost += lowestCost;
}
// console.log("totalCost: " + totalCost);
return list[0];
}
// TODO This should be improved using exponentiation by squaring or binary exponentiation, and something similar could be done for a power() method that raises a matrix to a given exponent
static tensorPower( matrix, exponent ) {
Util.assert( exponent>0 && matrix instanceof CMatrix, "CMatrix.naryTensor(): invalid input" );
let list = [];
for ( let i = 0; i < exponent; ++i ) {
list.push( matrix );
}
return CMatrix.naryTensor( list );
}
// If the caller needs to know Trace(A * B),
// this subroutine saves time by only computing the diagonal elements of the matrix product.
// This is equivalent to taking the transpose of one of the matrices,
// doing element-wise multiplication, and adding up all the elements in the resulting matrix.
//
static traceOfMatrixProduct( A, B ) {
Util.assert(A._cols === B._rows, "CMatrix.traceOfMatrixProduct(): matrices have incompatible dimensions" );
Util.assert(A._rows === B._cols, "CMatrix.traceOfMatrixProduct(): matrix product is not square, hence trace is not defined" );
let product_size = A._rows;
let returnValue = new Complex();
for ( let ii = 0; ii < product_size; ii++ ) {
let dotProduct = new Complex();
for ( let k = 0; k < A._cols; ++k ) {
// TODO this line creates 4 instances of Complex that ultimately aren't
// needed; this could be optimized by expanding and inlining the math ops.
dotProduct = Complex.sum(dotProduct,Complex.mult(A.get(ii,k),B.get(k,ii)));
}
returnValue = Complex.sum( returnValue, dotProduct );
}
return returnValue;
}
// returns true if the given matrices are approximately equal, within the given tolerance
static approximatelyEqual(a,b,tolerance=precisionForApproximateComparison,printMessage=true) {
Util.assert(a instanceof CMatrix && b instanceof CMatrix,"CMatrix.approximatelyEqual(): wrong type");
Util.assert(a._rows === b._rows && a._cols === b._cols, "CMatrix.approximatelyEqual(): incompatible dimensions" );
for ( let j = a._m.length-1; j >= 0; j-- ) {
let delta = Math.abs( a._m[j] - b._m[j] );
if ( delta > tolerance ) {
if ( printMessage ) {
console.log(`CMatrix.approximatelyEqual(): difference of ${delta} found`);
}
return false;
}
}
return true;
}
// Creates and returns a matrix by copying the contents of the given array of arrays,
// which can contain numbers or complex numbers.
// Assumes that the given array is not jagged.
static create( arrayOfArrays ) {
let numRows = arrayOfArrays.length;
let numCols = arrayOfArrays[0].length;
let M = new CMatrix( numRows, numCols );
for ( let j = 0; j < numRows; ++j ) {
for ( let k = 0; k < numCols; ++k ) {
M.set( j, k, arrayOfArrays[j][k] );
}
}
return M;
}
// Creates a nx1 size matrix
static createColVector( array ) {
return this.create( array.map( x => [ x ] ) );
}
// Creates a 1xn size matrix
static createRowVector( array ) {
return this.create( [ array ] );
}
// Returns an identity matrix of the given size
static identity( numRows ) {
let M = new CMatrix( numRows, numRows );
for ( let k = 0; k < numRows; ++k )
M.set(k,k,1);
return M;
}
}
class Sim { // Simulator
static GlobalPhase( angleInDegrees ) {
let angleInRadians = angleInDegrees / 180.0 * Math.PI;
let globalPhaseChange = new Complex( Math.cos(angleInRadians), Math.sin(angleInRadians) );
return CMatrix.create([[globalPhaseChange,0],[0,globalPhaseChange]]);
}
// Note that
// Phase(angle) * GlobalPhase( -angle/2 ) = RZ( angle )
// Phase(angle) = RZ( angle ) * GlobalPhase( angle/2 )
static Phase( angleInDegrees ) {
let angleInRadians = angleInDegrees / 180.0 * Math.PI;
let phaseChange = new Complex( Math.cos(angleInRadians), Math.sin(angleInRadians) );
return CMatrix.create([[1,0],[0,phaseChange]]);
}
// Returns a 2x2 matrix that rotates around an axis of the Bloch sphere.
// With a 180 degree angle, this is equivalent to a Pauli X gate (also called NOT gate)
// but differs by a global phase.
// More particularly, the following print statements should output the same thing:
// console.log(CMatrix.mult(Sim.RX(180),Sim.GlobalPhase(90)).toString());
// console.log(Sim.X.toString());
//
static RX( angleInDegrees ) {
let angleInRadians = angleInDegrees / 180.0 * Math.PI;
let halfAngle = angleInRadians / 2;
let sine = Math.sin( halfAngle );
let cosine = Math.cos( halfAngle );
return CMatrix.create( [
[ new Complex(cosine,0), new Complex(0,-sine) ],
[ new Complex(0,-sine), new Complex(cosine,0) ],
] );
}
// Returns a 2x2 matrix that rotates around an axis of the Bloch sphere.
// With a 180 degree angle, this is equivalent to a Pauli Y gate
// but differs by a global phase.
// More particularly, the following print statements should output the same thing:
// console.log(CMatrix.mult(Sim.RY(180),Sim.GlobalPhase(90)).toString());
// console.log(Sim.Y.toString());
//
static RY( angleInDegrees ) {
let angleInRadians = angleInDegrees / 180.0 * Math.PI;
let halfAngle = angleInRadians / 2;
let sine = Math.sin( halfAngle );
let cosine = Math.cos( halfAngle );
return CMatrix.create( [
[ new Complex(cosine,0), new Complex(-sine,0) ],
[ new Complex(sine,0), new Complex(cosine,0) ],
] );
}
// Returns a 2x2 matrix that rotates around an axis of the Bloch sphere.
// With a 180 degree angle, this is equivalent to a Pauli Z gate
// but differs by a global phase.
// More particularly, the following print statements should output the same thing:
// console.log(CMatrix.mult(Sim.RZ(180),Sim.GlobalPhase(90)).toString());
// console.log(Sim.Z.toString());
//
static RZ( angleInDegrees ) {
let angleInRadians = angleInDegrees / 180.0 * Math.PI;
let halfAngle = angleInRadians / 2;
let sine = Math.sin( halfAngle );
let cosine = Math.cos( halfAngle );
return CMatrix.create( [
[ new Complex(cosine,-sine), new Complex(0,0) ],
[ new Complex(0,0), new Complex(cosine,sine) ],
] );
}
// This rotates around the given vector by an angle in radians equal to the magnitude of the given vector.
// This way of encoding the angle in the vector of the axis of rotation might seem strange,
// but it means that the output is continuous in all 3 input variables and therefore easier to optimize,
// which would not be the case with a parametrization where the angle is separate from the vector components of the axis.
// (To see why, consider the latter parametrization, and imagine what would happen as the input shifts
// from (ax,ay-epsilon,az,angle) to (ax,ay+epsilon,az,angle).
// This would result in a 180 degree change of axis direction and a discontinuous change in output.
// The same kind of change of axis direction in the first parametrization would require the angle to pass through zero,
// avoiding a discontinuity in the output.)
//
// For the definition of the matrix, see
// https://arxiv.org/abs/2104.14875
// Hiroshi C. Watanabe, Rudy Raymond, Yu-ya Ohnishi, Eriko Kaminishi, Michihiko Sugawara
// Optimizing Parameterized Quantum Circuits with Free-Axis Selection
// in particular equation (1) on page 4.
// See also the documentation for the RVGate in Qiskit
// https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.RVGate
// although, at the time of writing, that documentation contains typos.
// See also the source code for the RVGate in Qiskit
// https://github.com/Qiskit/qiskit/blob/stable/0.46/qiskit/circuit/library/generalized_gates/rv.py
//
// If (ax,ay,az) is proportional to (1,0,0), or to (0,1,0), or to (0,0,1),
// the result is equivalent to RX, RY, or RZ, respectively.
static RotFreeAxis( ax, ay, az ) {
let magSquared = ax*ax + ay*ay + az*az;
if ( magSquared === 0 ) {
return Sim.I;
}
let angleInRadians = Math.sqrt( magSquared );
ax /= angleInRadians;
ay /= angleInRadians;
az /= angleInRadians;
let halfAngle = angleInRadians / 2;
let sine = Math.sin( halfAngle );
let cosine = Math.cos( halfAngle );
return CMatrix.create( [
[ new Complex(cosine,-az*sine), new Complex(-ay*sine,-ax*sine) ],
[ new Complex(ay*sine,-ax*sine), new Complex(cosine,az*sine) ],
] );
}
static RotFreeAxisAngle( ax, ay, az, angleInDegrees ) { // rotates around the given axis by the given angle
let angleInRadians = angleInDegrees / 180.0 * Math.PI;
let magSquared = ax*ax + ay*ay + az*az;
if ( magSquared === 0 || angleInDegrees === 0 ) {
return Sim.I;
}
let k = angleInRadians / Math.sqrt( magSquared );
return Sim.RotFreeAxis( ax * k, ay * k, az * k );
}
// Returns a matrix for swapping wires i and j in a quantum circuit with n wires.
// i and j are zero-based, i.e., they are between 0 and (n-1).
// The matrix that is returned has size (2**n)x(2**n)
static SWAP(i,j,n) {
Util.assert( 0<=i && i<n && 0<=j && j<n, "Sim.SWAP(): invalid indices" );
Util.assert( i!=j, "Sim.SWAP(): indices are unexpectedly equal" );
let matrixSize = 2 ** n;
if ( i === j ) {
return CMatrix.identity( matrixSize );
}
let M = new CMatrix( matrixSize, matrixSize );
if ( usingTextbookConvention ) {
// inverting the wire indices like this is equivalent to calling reverseEndianness() on the matrix just before returning it, but this is more efficient
i = n-1 - i;
j = n-1 - j;
}
// define stuff for operating on bits
let mask_i = 1 << i;
let mask_j = 1 << j;
let antimask = ~( mask_i | mask_j );
for ( let r = 0; r < matrixSize; ++r ) {
let c = r;
// BEGIN: swap the ith and jth bits of c
let extracted_bit_i = c & mask_i;
let extracted_bit_j = c & mask_j;
c &= antimask; // turns off bits i and j
if ( extracted_bit_i !== 0 ) c |= mask_j; // turns on bit j
if ( extracted_bit_j !== 0 ) c |= mask_i; // turns on bit i
// END
M.set(r,c,1);
}
return M;
}
// Given a 4x4 matrix, which operates on two qubits q0 and q1,
// this routine modifies it so that it will operate on the ith and jth qubits
// of an n-qubit circuit.
// The matrix that is returned has size (2**n)x(2**n)
// For example, passing in arguments (m,1,0,2)
// returns an 'upside-down' version of the original matrix.
static expand4x4ForNWires(m/*a 4x4 matrix*/,i,j,n/*number of qubits in the circuit*/) {
Util.assert( 0<=i && i<n && 0<=j && j<n && i!=j, "Sim.expand4x4ForNWires(): invalid indices" );
Util.assert( m._rows===m._cols && m._rows===4, "Sim.expand4x4ForNWires(): invalid size" );
// Step 1: turn the matrix upside down, if necessary
let m1 = null;
if ( i > j ) {
//
// +----+ +---+
// q_0 --| |-- q_0 --X--| |--X--
// | m1 | = | | m | |
// q_1 --| |-- q_1 --X--| |--X--
// +----+ +---+
//
// +----+ +---+
// q_i' = q_j --| |-- q_j --X--| |--X--
// | m1 | = | | m | |
// q_j' = q_i --| |-- q_i --X--| |--X--
// +----+ +---+
//
m1 = CMatrix.naryMult([ Sim.SWAP_2, m, Sim.SWAP_2 ]);
let tmp = i;
i = j;
j = tmp;
Util.assert(i<j,"Sim.expand4x4ForNWires(): unexpected condition");
}
else {
m1 = m.copy();
}
// Step 2: account for wires in between i and j
let m2 = null;
let numInnerWires = j-i-1;
if ( numInnerWires > 0 ) {
//
// q_0 --+----+-- q_0 -----+----+-----
// | | | m1 |
// q_1 --| |-- q_1 --X--+----+--X--
// | | | |
// ... --| |-- ... --|----------|--
// | m2 | = | |
// | | ...
// | | | |
// ... --| |-- ... --|----------|--
// | | | |
// q_{j-i} --+----+-- q_{j-i} --X----------X--
//
let swapStep = CMatrix.tensor( Sim.SWAP(0,numInnerWires,numInnerWires+1), Sim.I, usingTextbookConvention );
m2 = CMatrix.naryMult( [
swapStep,
CMatrix.tensor( CMatrix.identity(2**numInnerWires), m1, usingTextbookConvention ),
swapStep
] );
}
else {
m2 = m1;
}
// Step 3: if there are wires before i, and/or after j,
// we must perform a tensor product with an appropriately sized identity matrix,
// either before and/or after m2, respectively.
// Rather than do these tensor products separately here,
// we put the relevant matrices in a list and let naryTensor()
// perform the tensor products in the optimal associative order.
//
let listOfMatrices = [ m2 ];
let numWiresBefore = i;
if ( numWiresBefore > 0 ) {
listOfMatrices.push( CMatrix.identity(2**numWiresBefore) ); // add to end of list
}
let numWiresAfter = n-1-j;
if ( numWiresAfter > 0 ) {
listOfMatrices.unshift( CMatrix.identity(2**numWiresAfter) ); // insert at beginning of list
}
let m3 = CMatrix.naryTensor( listOfMatrices, usingTextbookConvention );
return m3;
}
// Returns a new state vector.
// Avoids computing an explicit matrix of size (2**n)x(2**n), saving a lot of memory and time.
//
// Imagine the state vector partitioned into contiguous blocks of equal size,
// with each block partitioned into two half blocks.
// The below code steps through the blocks, and within each block, for each i,
// it will take the ith amplitude of the first half block and the ith amplitude
// of the second half block, store those two amplitudes in a temporary vector
// (this is the "collect" step, since it collects the amplitudes into the temporary vector),
// apply the 2x2 matrix to that temporary vector, and store the resulting amplitudes
// back in the ith positions of the half blocks (the "scatter" step).
//
// The below code is similar to the code in Quirk ( https://github.com/Strilanc/Quirk/ ,
// src/math/Matrix.js , applyToStateVectorAtQubitWithControls() )
// and also similar to the pseudocode in Figure 6.1 of the book by Viamontes et al.
// (Viamontes, G. F., Markov, I. L., & Hayes, J. P. (2009) "Quantum circuit simulation")
// Viamontes et al. call this algorithm "qubit-wise multiplication".
//
// To compare with Quirk's version of the code, it helps to know that Quirk's source code
// calls the tempVector a 'chunk', and it uses different names for several variables,
// listed below. Also, our version has indices and sizes of things in complex numbers
// (i.e., in amplitudes), whereas Quirk's corresponding quantities are in floats and
// are therefore doubled.
//
// Some quantities in our code, and the corresponding quantities in Quirk's code:
// sizeOfHalfBlock (in complex numbers) is (strideLength (in floats))/2
// sizeOfBlock (in complex numbers) is (strideChunkSize (in floats))/2
// indexOfStartOfBlock (in complex numbers) is (strideChunkStart (in floats))/2
// offsetWithinBlock (in complex numbers) is (strideOffset (in floats))/2
//
// Viamontes et al., Figure 6.1, appears to be the same algorithm. The inner "for" loop
// in it performs the "collect" step, and the line of code immediately following,
// which uses slice notation (res[a:...:gap]), performs the "scatter" step.
// The 'gap' variable in Viamontes et al.'s version is sizeOfHalfBlock in our code.
// However, Viamontes et al., Figure 6.1 also contains errors: it mentions a
// "group_factor" that is undefined, and contains a stray closing brace.
// There are other things about it that are confusing which I haven't
// fully analyzed. Finally, Viamontes et al.'s version has no support for control bits.
//
static transformStateVectorWith2x2(
m, // a 2x2 matrix
i, // index of wire on which to apply m
n, // number of qubits in the circuit
stateVector, // a matrix of size (2**n)x1, i.e. a column vector
listOfControlBits = [] // a list of pairs of the form [wire_index, flag] where 0<=wire_index<n and flag is true for a control bit and false for an anti-control bit
) {
Util.assert( 0<=i && i<n, "Sim.transformStateVectorWith2x2(): invalid index" );
Util.assert( m._rows===m._cols && m._rows===2, "Sim.transformStateVectorWith2x2(): invalid size" );
Util.assert( stateVector._rows===(2**n) && stateVector._cols===1, "Sim.transformStateVectorWith2x2(): state vector has invalid size" );
let inclusionMask = 0;
let desiredValueMask = 0;
for ( let iter of listOfControlBits ) {
let [wireIndex,flag] = iter;
if ( usingTextbookConvention )
wireIndex = n-1 - wireIndex;
let bit = 1 << wireIndex;
inclusionMask |= bit;
if ( flag )
desiredValueMask |= bit;
}
if ( usingTextbookConvention )
i = n-1 - i;
let tempVector = new CMatrix(2,1);
let sizeOfHalfBlock = 1 << i; // could be 1, 2, 4, ...
let sizeOfBlock = sizeOfHalfBlock * 2; // could be 2, 4, 8 ...
let result = stateVector.copy();
for ( let indexOfStartOfBlock = 0; indexOfStartOfBlock < stateVector._rows; indexOfStartOfBlock += sizeOfBlock ) {
for ( let offsetWithinBlock = 0; offsetWithinBlock < sizeOfHalfBlock; offsetWithinBlock ++ ) {
Util.assert( indexOfStartOfBlock | offsetWithinBlock === indexOfStartOfBlock + offsetWithinBlock, "Sim.transformStateVectorWith2x2(): unexpected numbers" );
let stateIndex1 = indexOfStartOfBlock | offsetWithinBlock;
if ( (stateIndex1 & inclusionMask) !== desiredValueMask )
continue;
let stateIndex2 = stateIndex1 + sizeOfHalfBlock;
// Collect inputs into a small contiguous vector.
tempVector.set( 0, 0, stateVector.get( stateIndex1, 0 ) );
tempVector.set( 1, 0, stateVector.get( stateIndex2, 0 ) );
let transformedVector = CMatrix.mult( m, tempVector );
// Scatter outputs.
result.set( stateIndex1, 0, transformedVector.get( 0, 0 ) );
result.set( stateIndex2, 0, transformedVector.get( 1, 0 ) );
}
}
return result;
}
// Returns a new state vector.
// Avoids computing an explicit swap matrix of size (2**n)x(2**n), saving a lot of memory and time.
static transformStateVectorWithSwap(
i, // index of 1st wire on which to apply the swap
j, // index of 2nd wire on which to apply the swap
n, // number of qubits in the circuit
stateVector, // a matrix of size (2**n)x1, i.e. a column vector
listOfControlBits = [] // a list of pairs of the form [wire_index, flag] where 0<=wire_index<n and flag is true for a control bit and false for an anti-control bit
) {
Util.assert( 0<=i && i<n && 0<=j && j<n, "Sim.transformStateVectorWithSwap(): invalid indices" );
Util.assert( i!=j, "Sim.transformStateVectorWithSwap(): indices are unexpectedly equal" );
Util.assert( stateVector._rows===(2**n) && stateVector._cols===1, "Sim.transformStateVectorWithSwap(): state vector has invalid size" );
let result = stateVector.copy();