Expression of type Conditional ¶

from the theory of proveit.physics.quantum.QPE ¶

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, ExprRange, U, Variable, VertExprArray, s, t
from proveit.linear_algebra import MatrixMult, ScalarMult
from proveit.logic import And, Equals, InSet
from proveit.numbers import Add, Exp, Interval, IntervalCO, Mod, Mult, Real, Round, e, four, frac, greater, i, one, pi, two, zero
from proveit.physics.quantum import I, NumKet, Z, ket_plus, var_ket_u
from proveit.physics.quantum.QPE import QPE, phase, two_pow_t
from proveit.physics.quantum.circuits import Gate, Input, Measure, MultiQubitElem, Output, Qcircuit
from proveit.statistics import Prob

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(t, one)
sub_expr3 = Add(t, s)
sub_expr4 = Interval(sub_expr2, sub_expr3)
sub_expr5 = MultiQubitElem(element = Gate(operation = QPE(U, t), part = sub_expr1), targets = Interval(one, sub_expr3))
expr = Conditional(greater(Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Input(state = ket_plus), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = var_ket_u, part = sub_expr1), targets = sub_expr4), one, s)], [ExprRange(sub_expr1, sub_expr5, one, t), ExprRange(sub_expr1, sub_expr5, sub_expr2, sub_expr3)], [ExprRange(sub_expr1, Measure(basis = Z), one, t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, s)], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(Mod(Round(Mult(two_pow_t, phase)), two_pow_t), t), part = sub_expr1), targets = Interval(one, t)), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = var_ket_u, part = sub_expr1), targets = sub_expr4), one, s)]))), frac(four, Exp(pi, two))), And(InSet(phase, Real), InSet(phase, IntervalCO(zero, one)), Equals(MatrixMult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, phase)), var_ket_u))))

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left\{\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert round\left(2^{t} \cdot \varphi\right) ~\textup{mod}~ 2^{t} \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) > \frac{4}{\pi^{2}} \textrm{ if } \varphi \in \mathbb{R} ,  \varphi \in \left[0,1\right) ,  \left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right)\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operands: 4
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7, 8
5	Literal
6	ExprTuple	9, 10, 11
7	Operation	operator: 12 operands: 13
8	Operation	operator: 14 operand: 23
9	Operation	operator: 17 operands: 16
10	Operation	operator: 17 operands: 18
11	Operation	operator: 19 operands: 20
12	Literal
13	ExprTuple	21, 22
14	Literal
15	ExprTuple	23
16	ExprTuple	130, 24
17	Literal
18	ExprTuple	130, 25
19	Literal
20	ExprTuple	26, 27
21	Literal
22	Operation	operator: 131 operands: 28
23	Operation	operator: 29 operands: 30
24	Literal
25	Operation	operator: 31 operands: 32
26	Operation	operator: 33 operands: 34
27	Operation	operator: 35 operands: 36
28	ExprTuple	79, 133
29	Literal
30	ExprTuple	37, 38, 39, 40
31	Literal
32	ExprTuple	41, 119
33	Literal
34	ExprTuple	117, 106
35	Literal
36	ExprTuple	42, 106
37	ExprTuple	43, 44
38	ExprTuple	45, 46
39	ExprTuple	47, 48
40	ExprTuple	49, 50
41	Literal
42	Operation	operator: 131 operands: 51
43	ExprRange	lambda_map: 52 start_index: 119 end_index: 134
44	ExprRange	lambda_map: 53 start_index: 119 end_index: 120
45	ExprRange	lambda_map: 54 start_index: 119 end_index: 134
46	ExprRange	lambda_map: 54 start_index: 108 end_index: 109
47	ExprRange	lambda_map: 55 start_index: 119 end_index: 134
48	ExprRange	lambda_map: 56 start_index: 119 end_index: 120
49	ExprRange	lambda_map: 57 start_index: 119 end_index: 134
50	ExprRange	lambda_map: 58 start_index: 119 end_index: 120
51	ExprTuple	59, 60
52	Lambda	parameter: 107 body: 61
53	Lambda	parameter: 107 body: 62
54	Lambda	parameter: 107 body: 63
55	Lambda	parameter: 107 body: 64
56	Lambda	parameter: 107 body: 65
57	Lambda	parameter: 107 body: 66
58	Lambda	parameter: 107 body: 68
59	Literal
60	Operation	operator: 127 operands: 69
61	Operation	operator: 93 operands: 70
62	Operation	operator: 77 operands: 71
63	Operation	operator: 77 operands: 72
64	Operation	operator: 73 operands: 74
65	Operation	operator: 94 operands: 75
66	Operation	operator: 77 operands: 76
67	ExprTuple	107
68	Operation	operator: 77 operands: 78
69	ExprTuple	133, 79, 80, 130
70	NamedExprs	state: 81
71	NamedExprs	element: 82 targets: 90
72	NamedExprs	element: 83 targets: 84
73	Literal
74	NamedExprs	basis: 85
75	NamedExprs	operation: 86
76	NamedExprs	element: 87 targets: 88
77	Literal
78	NamedExprs	element: 89 targets: 90
79	Literal
80	Literal
81	Operation	operator: 91 operand: 103
82	Operation	operator: 93 operands: 100
83	Operation	operator: 94 operands: 95
84	Operation	operator: 101 operands: 96
85	Literal
86	Literal
87	Operation	operator: 99 operands: 97
88	Operation	operator: 101 operands: 98
89	Operation	operator: 99 operands: 100
90	Operation	operator: 101 operands: 102
91	Literal
92	ExprTuple	103
93	Literal
94	Literal
95	NamedExprs	operation: 104 part: 107
96	ExprTuple	119, 109
97	NamedExprs	state: 105 part: 107
98	ExprTuple	119, 134
99	Literal
100	NamedExprs	state: 106 part: 107
101	Literal
102	ExprTuple	108, 109
103	Literal
104	Operation	operator: 110 operands: 111
105	Operation	operator: 112 operands: 113
106	Variable
107	Variable
108	Operation	operator: 115 operands: 114
109	Operation	operator: 115 operands: 116
110	Literal
111	ExprTuple	117, 134
112	Literal
113	ExprTuple	118, 134
114	ExprTuple	134, 119
115	Literal
116	ExprTuple	134, 120
117	Variable
118	Operation	operator: 121 operands: 122
119	Literal
120	Variable
121	Literal
122	ExprTuple	123, 129
123	Operation	operator: 124 operand: 126
124	Literal
125	ExprTuple	126
126	Operation	operator: 127 operands: 128
127	Literal
128	ExprTuple	129, 130
129	Operation	operator: 131 operands: 132
130	Variable
131	Literal
132	ExprTuple	133, 134
133	Literal
134	Variable

Expression of type Conditional¶

from the theory of proveit.physics.quantum.QPE¶

Expression of type Conditional ¶

from the theory of proveit.physics.quantum.QPE ¶