// CHP: CNOT-Hadamard-Phase // Stabilizer Quantum Computer Simulator // by Scott Aaronson // Last modified June 30, 2004 // Thanks to Simon Anders and Andrew Cross for bugfixes #include #include #include #include #include #define CNOT 0 #define HADAMARD 1 #define PHASE 2 #define MEASURE 3 struct QProg // Quantum circuit { long n; // # of qubits long T; // # of gates char *a; // Instruction opcode long *b; // Qubit 1 long *c; // Qubit 2 (target for CNOT) int DISPQSTATE; // whether to print the state (q for final state only, Q for every iteration) int DISPTIME; // whether to print the execution time int SILENT; // whether NOT to print measurement results int DISPPROG; // whether to print instructions being executed as they're executed int SUPPRESSM; // whether to suppress actual computation of determinate measurement results }; struct QState // Quantum state { // To save memory and increase speed, the bits are packed 32 to an unsigned long long n; // # of qubits unsigned long **x; // (2n+1)*n matrix for stabilizer/destabilizer x bits (there's one "scratch row" at unsigned long **z; // (2n+1)*n matrix for z bits the bottom) int *r; // Phase bits: 0 for +1, 1 for i, 2 for -1, 3 for -i. Normally either 0 or 2. unsigned long pw[32]; // pw[i] = 2^i long over32; // floor(n/8)+1 }; // Globals struct QProg *h; struct QState *q; void error(int k) { if (k==0) printf("\nSyntax: chp [-options] [input]\n"); if (k==1) printf("\nFile not found\n"); exit(0); } void cnot(struct QState *q, long b, long c) // Apply a CNOT gate with control b and target c { long i; long b5; long c5; unsigned long pwb; unsigned long pwc; b5 = b>>5; c5 = c>>5; pwb = q->pw[b&31]; pwc = q->pw[c&31]; for (i = 0; i < 2*q->n; i++) { if (q->x[i][b5]&pwb) q->x[i][c5] ^= pwc; if (q->z[i][c5]&pwc) q->z[i][b5] ^= pwb; if ((q->x[i][b5]&pwb) && (q->z[i][c5]&pwc) && (q->x[i][c5]&pwc) && (q->z[i][b5]&pwb)) q->r[i] = (q->r[i]+2)%4; if ((q->x[i][b5]&pwb) && (q->z[i][c5]&pwc) && !(q->x[i][c5]&pwc) && !(q->z[i][b5]&pwb)) q->r[i] = (q->r[i]+2)%4; } return; } void hadamard(struct QState *q, long b) // Apply a Hadamard gate to qubit b { long i; unsigned long tmp; long b5; unsigned long pw; b5 = b>>5; pw = q->pw[b&31]; for (i = 0; i < 2*q->n; i++) { tmp = q->x[i][b5]; q->x[i][b5] ^= (q->x[i][b5] ^ q->z[i][b5]) & pw; q->z[i][b5] ^= (q->z[i][b5] ^ tmp) & pw; if ((q->x[i][b5]&pw) && (q->z[i][b5]&pw)) q->r[i] = (q->r[i]+2)%4; } return; } void phase(struct QState *q, long b) // Apply a phase gate (|0>->|0>, |1>->i|1>) to qubit b { long i; long b5; unsigned long pw; b5 = b>>5; pw = q->pw[b&31]; for (i = 0; i < 2*q->n; i++) { if ((q->x[i][b5]&pw) && (q->z[i][b5]&pw)) q->r[i] = (q->r[i]+2)%4; q->z[i][b5] ^= q->x[i][b5]&pw; } return; } void rowcopy(struct QState *q, long i, long k) // Sets row i equal to row k { long j; for (j = 0; j < q->over32; j++) { q->x[i][j] = q->x[k][j]; q->z[i][j] = q->z[k][j]; } q->r[i] = q->r[k]; return; } void rowswap(struct QState *q, long i, long k) // Swaps row i and row k { rowcopy(q, 2*q->n, k); rowcopy(q, k, i); rowcopy(q, i, 2*q->n); return; } void rowset(struct QState *q, long i, long b) // Sets row i equal to the bth observable (X_1,...X_n,Z_1,...,Z_n) { long j; long b5; unsigned long b31; for (j = 0; j < q->over32; j++) { q->x[i][j] = 0; q->z[i][j] = 0; } q->r[i] = 0; if (b < q->n) { b5 = b>>5; b31 = b&31; q->x[i][b5] = q->pw[b31]; } else { b5 = (b - q->n)>>5; b31 = (b - q->n)&31; q->z[i][b5] = q->pw[b31]; } return; } int clifford(struct QState *q, long i, long k) // Return the phase (0,1,2,3) when row i is LEFT-multiplied by row k { long j; long l; unsigned long pw; long e=0; // Power to which i is raised for (j = 0; j < q->over32; j++) for (l = 0; l < 32; l++) { pw = q->pw[l]; if ((q->x[k][j]&pw) && (!(q->z[k][j]&pw))) // X { if ((q->x[i][j]&pw) && (q->z[i][j]&pw)) e++; // XY=iZ if ((!(q->x[i][j]&pw)) && (q->z[i][j]&pw)) e--; // XZ=-iY } if ((q->x[k][j]&pw) && (q->z[k][j]&pw)) // Y { if ((!(q->x[i][j]&pw)) && (q->z[i][j]&pw)) e++; // YZ=iX if ((q->x[i][j]&pw) && (!(q->z[i][j]&pw))) e--; // YX=-iZ } if ((!(q->x[k][j]&pw)) && (q->z[k][j]&pw)) // Z { if ((q->x[i][j]&pw) && (!(q->z[i][j]&pw))) e++; // ZX=iY if ((q->x[i][j]&pw) && (q->z[i][j]&pw)) e--; // ZY=-iX } } e = (e+q->r[i]+q->r[k])%4; if (e>=0) return e; else return e+4; } void rowmult(struct QState *q, long i, long k) // Left-multiply row i by row k { long j; q->r[i] = clifford(q,i,k); for (j = 0; j < q->over32; j++) { q->x[i][j] ^= q->x[k][j]; q->z[i][j] ^= q->z[k][j]; } return; } void printstate(struct QState *q) // Print the destabilizer and stabilizer for state q { long i; long j; long j5; unsigned long pw; for (i = 0; i < 2*q->n; i++) { if (i == q->n) { printf("\n"); for (j = 0; j < q->n+1; j++) printf("-"); } if (q->r[i]==2) printf("\n-"); else printf("\n+"); for (j = 0; j < q->n; j++) { j5 = j>>5; pw = q->pw[j&31]; if ((!(q->x[i][j5]&pw)) && (!(q->z[i][j5]&pw))) printf("I"); if ((q->x[i][j5]&pw) && (!(q->z[i][j5]&pw))) printf("X"); if ((q->x[i][j5]&pw) && (q->z[i][j5]&pw)) printf("Y"); if ((!(q->x[i][j5]&pw)) && (q->z[i][j5]&pw)) printf("Z"); } } printf("\n"); return; } int measure(struct QState *q, long b, int sup) // Measure qubit b // Return 0 if outcome would always be 0 // 1 if outcome would always be 1 // 2 if outcome was random and 0 was chosen // 3 if outcome was random and 1 was chosen // sup: 1 if determinate measurement results should be suppressed, 0 otherwise { int ran = 0; long i; long p; // pivot row in stabilizer long m; // pivot row in destabilizer long b5; unsigned long pw; b5 = b>>5; pw = q->pw[b&31]; for (p = 0; p < q->n; p++) // loop over stabilizer generators { if (q->x[p+q->n][b5]&pw) ran = 1; // if a Zbar does NOT commute with Z_b (the if (ran) break; // operator being measured), then outcome is random } // If outcome is indeterminate if (ran) { rowcopy(q, p, p + q->n); // Set Xbar_p := Zbar_p rowset(q, p + q->n, b + q->n); // Set Zbar_p := Z_b q->r[p + q->n] = 2*(rand()%2); // moment of quantum randomness for (i = 0; i < 2*q->n; i++) // Now update the Xbar's and Zbar's that don't commute with if ((i!=p) && (q->x[i][b5]&pw)) // Z_b rowmult(q, i, p); if (q->r[p + q->n]) return 3; else return 2; } // If outcome is determinate if ((!ran) && (!sup)) { for (m = 0; m < q->n; m++) // Before we were checking if stabilizer generators commute if (q->x[m][b5]&pw) break; // with Z_b; now we're checking destabilizer generators rowcopy(q, 2*q->n, m + q->n); for (i = m+1; i < q->n; i++) if (q->x[i][b5]&pw) rowmult(q, 2*q->n, i + q->n); if (q->r[2*q->n]) return 1; else return 0; /*for (i = m+1; i < q->n; i++) if (q->x[i][b5]&pw) { rowmult(q, m + q->n, i + q->n); rowmult(q, i, m); } return (int)q->r[m + q->n];*/ } return 0; } long gaussian(struct QState *q) // Do Gaussian elimination to put the stabilizer generators in the following form: // At the top, a minimal set of generators containing X's and Y's, in "quasi-upper-triangular" form. // (Return value = number of such generators = log_2 of number of nonzero basis states) // At the bottom, generators containing Z's only in quasi-upper-triangular form. { long i = q->n; long k; long k2; long j; long j5; long g; // Return value unsigned long pw; for (j = 0; j < q->n; j++) { j5 = j>>5; pw = q->pw[j&31]; for (k = i; k < 2*q->n; k++) // Find a generator containing X in jth column if (q->x[k][j5]&pw) break; if (k < 2*q->n) { rowswap(q, i, k); rowswap(q, i-q->n, k-q->n); for (k2 = i+1; k2 < 2*q->n; k2++) if (q->x[k2][j5]&pw) { rowmult(q, k2, i); // Gaussian elimination step rowmult(q, i-q->n, k2-q->n); } i++; } } g = i - q->n; for (j = 0; j < q->n; j++) { j5 = j>>5; pw = q->pw[j&31]; for (k = i; k < 2*q->n; k++) // Find a generator containing Z in jth column if (q->z[k][j5]&pw) break; if (k < 2*q->n) { rowswap(q, i, k); rowswap(q, i-q->n, k-q->n); for (k2 = i+1; k2 < 2*q->n; k2++) if (q->z[k2][j5]&pw) { rowmult(q, k2, i); rowmult(q, i-q->n, k2-q->n); } i++; } } return g; } long innerprod(struct QState *q1, struct QState *q2) // Returns -1 if q1 and q2 are orthogonal // Otherwise, returns a nonnegative integer s such that the inner product is (1/sqrt(2))^s { return 0; } void printbasisstate(struct QState *q) // Prints the result of applying the Pauli operator in the "scratch space" of q to |0...0> { long j; long j5; unsigned long pw; int e = q->r[2*q->n]; for (j = 0; j < q->n; j++) { j5 = j>>5; pw = q->pw[j&31]; if ((q->x[2*q->n][j5]&pw) && (q->z[2*q->n][j5]&pw)) // Pauli operator is "Y" e = (e+1)%4; } if (e==0) printf("\n +|"); if (e==1) printf("\n+i|"); if (e==2) printf("\n -|"); if (e==3) printf("\n-i|"); for (j = 0; j < q->n; j++) { j5 = j>>5; pw = q->pw[j&31]; if (q->x[2*q->n][j5]&pw) printf("1"); else printf("0"); } printf(">"); return; } void dumpbasisstate(struct QState *q, PyObject *output, int index) // Prints the result of applying the Pauli operator in the "scratch space" of q to |0...0> { long j; long j5; unsigned long pw; int e = q->r[2*q->n]; char buffer[1024]; strcpy(buffer, ""); for (j = 0; j < q->n; j++) { j5 = j>>5; pw = q->pw[j&31]; if ((q->x[2*q->n][j5]&pw) && (q->z[2*q->n][j5]&pw)) // Pauli operator is "Y" e = (e+1)%4; } if (e==0) strcat(buffer, "+|"); if (e==1) strcat(buffer, "+i|"); if (e==2) strcat(buffer, "-|"); if (e==3) strcat(buffer, "-i|"); for (j = 0; j < q->n; j++) { j5 = j>>5; pw = q->pw[j&31]; if (q->x[2*q->n][j5]&pw) strcat(buffer, "1"); else strcat(buffer, "0"); } strcat(buffer, ">"); PyList_SetItem(output, index, Py_BuildValue("s", buffer)); return; } void seed(struct QState *q, long g) // Finds a Pauli operator P such that the basis state P|0...0> occurs with nonzero amplitude in q, and // writes P to the scratch space of q. For this to work, Gaussian elimination must already have been // performed on q. g is the return value from gaussian(q). { long i; long j; long j5; unsigned long pw; int f; long min=0; q->r[2*q->n] = 0; for (j = 0; j < q->over32; j++) { q->x[2*q->n][j] = 0; // Wipe the scratch space clean q->z[2*q->n][j] = 0; } for (i = 2*q->n - 1; i >= q->n + g; i--) { f = q->r[i]; for (j = q->n - 1; j >= 0; j--) { j5 = j>>5; pw = q->pw[j&31]; if (q->z[i][j5]&pw) { min = j; if (q->x[2*q->n][j5]&pw) f = (f+2)%4; } } if (f==2) { j5 = min>>5; pw = q->pw[min&31]; q->x[2*q->n][j5] ^= pw; // Make the seed consistent with the ith equation } } return; } static char act_hadamard_docs[] = "act_hadamard(): Do a Hadamard"; static PyObject* act_hadamard(PyObject* self, PyObject* args) { int which; if (!PyArg_ParseTuple(args, "i", &which)) { return NULL; } hadamard(q, which); Py_INCREF(Py_None); return Py_None; } static char act_phase_docs[] = "act_phase(): Do a phase"; static PyObject* act_phase(PyObject* self, PyObject* args) { int which; if (!PyArg_ParseTuple(args, "i", &which)) { return NULL; } phase(q, which); Py_INCREF(Py_None); return Py_None; } static char act_cnot_docs[] = "act_cnot(): Do a CNOT"; static PyObject* act_cnot(PyObject* self, PyObject* args) { int a, b; if (!PyArg_ParseTuple(args, "ii", &a, &b)) { return NULL; } cnot(q, a, b); Py_INCREF(Py_None); return Py_None; } static char get_ket_docs[] = "get_ket(): Get the state vector"; static PyObject* get_ket(PyObject* self, PyObject* noarg) // Print the state in ket notation (warning: could be huge!) { long g; // log_2 of number of nonzero basis states unsigned long t; unsigned long t2; long i; g = gaussian(q); PyObject *output = PyList_New(g+1); if (g > 30) { return output; } seed(q, g); dumpbasisstate(q, output, 0); for (t = 0; t < q->pw[g]-1; t++) { t2 = t ^ (t+1); for (i = 0; i < g; i++) if (t2 & q->pw[i]) rowmult(q, 2*q->n, q->n + i); dumpbasisstate(q, output, t+1); } return output; } void printket(struct QState *q) // Print the state in ket notation (warning: could be huge!) { long g; // log_2 of number of nonzero basis states unsigned long t; unsigned long t2; long i; g = gaussian(q); printf("\n2^%ld nonzero basis states", g); if (g > 31) { printf("\nState is WAY too big to print"); return; } seed(q, g); printbasisstate(q); for (t = 0; t < q->pw[g]-1; t++) { t2 = t ^ (t+1); for (i = 0; i < g; i++) if (t2 & q->pw[i]) rowmult(q, 2*q->n, q->n + i); printbasisstate(q); } printf("\n"); return; } void runprog(struct QProg *h, struct QState *q) // Simulate the quantum circuit { long t; int m; // measurement result time_t tp; double dt; char mvirgin = 1; time(&tp); for (t = 0; t < h->T; t++) { if (h->a[t]==CNOT) cnot(q,h->b[t],h->c[t]); if (h->a[t]==HADAMARD) hadamard(q,h->b[t]); if (h->a[t]==PHASE) phase(q,h->b[t]); if (h->a[t]==MEASURE) { if (mvirgin && h->DISPTIME) { dt = difftime(time(0),tp); printf("\nGate time: %lf seconds", dt); printf("\nTime per 10000 gates: %lf seconds", dt*10000.0f/(h->T - h->n)); time(&tp); } mvirgin = 0; m = measure(q,h->b[t],h->SUPPRESSM); if (!h->SILENT) { printf("\nOutcome of measuring qubit %ld: ", h->b[t]); if (m>1) printf("%d (random)", m-2); else printf("%d", m); } } if (h->DISPPROG) { if (h->a[t]==CNOT) printf("\nCNOT %ld->%ld", h->b[t], h->c[t]); if (h->a[t]==HADAMARD) printf("\nHadamard %ld", h->b[t]); if (h->a[t]==PHASE) printf("\nPhase %ld", h->b[t]); } } printf("\n"); if (h->DISPTIME) { dt = difftime(time(0),tp); printf("\nMeasurement time: %lf seconds", dt); printf("\nTime per 10000 measurements: %lf seconds\n", dt*10000.0f/h->n); } if (h->DISPQSTATE) { printf("\nFinal state:"); printstate(q); gaussian(q); printstate(q); printket(q); } return; } void preparestate(struct QState *q, char *s) // Prepare the initial state's "input" { long l; long b; l = strlen(s); for (b = 0; b < l; b++) { if (s[b]=='Z') { hadamard(q,b); phase(q,b); phase(q,b); hadamard(q,b); } if (s[b]=='x') hadamard(q,b); if (s[b]=='X') { hadamard(q,b); phase(q,b); phase(q,b); } if (s[b]=='y') { hadamard(q,b); phase(q,b); } if (s[b]=='Y') { hadamard(q,b); phase(q,b); phase(q,b); phase(q,b); } } return; } void initstae_(struct QState *q, long n, char *s) // Initialize state q to have n qubits, and input specified by s { long i; long j; q->n = n; q->x = malloc((2*q->n + 1) * sizeof(unsigned long*)); q->z = malloc((2*q->n + 1) * sizeof(unsigned long*)); q->r = malloc((2*q->n + 1) * sizeof(int)); q->over32 = (q->n>>5) + 1; q->pw[0] = 1; for (i = 1; i < 32; i++) q->pw[i] = 2*q->pw[i-1]; for (i = 0; i < 2*q->n + 1; i++) { q->x[i] = malloc(q->over32 * sizeof(unsigned long)); q->z[i] = malloc(q->over32 * sizeof(unsigned long)); for (j = 0; j < q->over32; j++) { q->x[i][j] = 0; q->z[i][j] = 0; } if (i < q->n) q->x[i][i>>5] = q->pw[i&31]; else if (i < 2*q->n) { j = i-q->n; q->z[i][j>>5] = q->pw[j&31]; } q->r[i] = 0; } if (s) preparestate(q, s); return; } void readprog(struct QProg *h, char *fn, char *params) // Read a quantum circuit from filename fn, with optional parameters params { long t; char fn2[255]; FILE *fp; char c=0; long val; long l; h->DISPQSTATE = 0; h->DISPTIME = 0; h->SILENT = 0; h->DISPPROG = 0; h->SUPPRESSM = 0; if (params) { l = strlen(params); for (t = 1; t < l; t++) { if ((params[t]=='q')||(params[t]=='Q')) h->DISPQSTATE = 1; if ((params[t]=='p')||(params[t]=='P')) h->DISPPROG = 1; if ((params[t]=='t')||(params[t]=='T')) h->DISPTIME = 1; if ((params[t]=='s')||(params[t]=='S')) h->SILENT = 1; if ((params[t]=='m')||(params[t]=='M')) h->SUPPRESSM = 1; } } sprintf(fn2, "%s", fn); fp = fopen(fn2, "r"); if (!fp) { sprintf(fn2, "%s.chp", fn); fp = fopen(fn2, "r"); if (!fp) error(1); } while (!feof(fp)&&(c!='#')) fscanf(fp, "%c", &c); if (c!='#') error(2); h->T = 0; h->n = 0; while (!feof(fp)) { fscanf(fp, "%c", &c); if ((c=='\r')||(c=='\n')) continue; fscanf(fp, "%ld", &val); if (val+1 > h->n) h->n = val+1; if ((c=='c')||(c=='C')) { fscanf(fp, "%ld", &val); if (val+1 > h->n) h->n = val+1; } h->T++; } fclose(fp); h->a = malloc(h->T * sizeof(char)); h->b = malloc(h->T * sizeof(long)); h->c = malloc(h->T * sizeof(long)); fp = fopen(fn2, "r"); while (!feof(fp)&&(c!='#')) fscanf(fp, "%c", &c); t=0; while (!feof(fp)) { fscanf(fp, "%c", &c); if ((c=='\r')||(c=='\n')) continue; if ((c=='c')||(c=='C')) h->a[t] = CNOT; if ((c=='h')||(c=='H')) h->a[t] = HADAMARD; if ((c=='p')||(c=='P')) h->a[t] = PHASE; if ((c=='m')||(c=='M')) h->a[t] = MEASURE; fscanf(fp, "%ld", &h->b[t]); if (h->a[t]==CNOT) fscanf(fp, "%ld", &h->c[t]); t++; } fclose(fp); return; } static char init_docs[] = "init(nqubits): Initialize with n qubits"; static PyObject* init(PyObject* self, PyObject* args) { int nqubits; if (!PyArg_ParseTuple(args, "i", &nqubits)) { return NULL; } srand(time(0)); h = malloc(sizeof(struct QProg)); q = malloc(sizeof(struct QState)); initstae_(q,nqubits,NULL); PyObject *response = Py_BuildValue("i", nqubits); return response; } static PyMethodDef chp_funcs[] = { {"init", (PyCFunction)init, METH_VARARGS, init_docs}, {"get_ket", (PyCFunction)get_ket, METH_NOARGS, get_ket_docs}, {"act_hadamard", (PyCFunction)act_hadamard, METH_VARARGS, act_hadamard_docs}, {"act_cnot", (PyCFunction)act_cnot, METH_VARARGS, act_cnot_docs}, {"act_phase", (PyCFunction)act_phase, METH_VARARGS, act_phase_docs}, {NULL} }; void initchp(int argc, char **argv) { Py_InitModule3("chp", chp_funcs, "CHP"); }