Expression of type Lambda

from the theory of proveit.physics.quantum.circuits

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, Lambda, U
from proveit.linear_algebra import ScalarMult
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, e, i, one, pi, two
from proveit.physics.quantum import Qmult, var_ket_u, varphi
from proveit.physics.quantum.circuits import phase_kickback_circuit
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Lambda(varphi, Conditional(Equals(Prob(phase_kickback_circuit), one), Equals(Qmult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, varphi)), var_ket_u))))

expr:

# 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())

\varphi \mapsto \left\{\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \control{} \qw \qwx[1] & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert u \rangle} & \gate{U} & \qout{\lvert u \rangle}
} \end{array}\right) = 1 \textrm{ if } \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 101 body: 2
1	ExprTuple	101
2	Conditional	value: 3 condition: 4
3	Operation	operator: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 95
6	Literal
7	ExprTuple	9, 10
8	Operation	operator: 11 operand: 16
9	Operation	operator: 13 operands: 14
10	Operation	operator: 78 operands: 15
11	Literal
12	ExprTuple	16
13	Literal
14	ExprTuple	69, 73
15	ExprTuple	84, 73
16	Operation	operator: 17 operands: 18
17	Literal
18	ExprTuple	19, 20, 21
19	ExprTuple	22, 23
20	ExprTuple	24, 25
21	ExprTuple	26, 27
22	Operation	operator: 59 operands: 28
23	ExprRange	lambda_map: 29 start_index: 95 end_index: 34
24	Operation	operator: 50 operands: 30
25	ExprRange	lambda_map: 31 start_index: 95 end_index: 34
26	Operation	operator: 65 operands: 32
27	ExprRange	lambda_map: 33 start_index: 95 end_index: 34
28	NamedExprs	state: 35
29	Lambda	parameter: 74 body: 36
30	NamedExprs	element: 37 targets: 38
31	Lambda	parameter: 74 body: 39
32	NamedExprs	state: 40
33	Lambda	parameter: 74 body: 42
34	Operation	operator: 80 operands: 43
35	Operation	operator: 91 operand: 52
36	Operation	operator: 50 operands: 45
37	Literal
38	Operation	operator: 46 operand: 98
39	Operation	operator: 50 operands: 48
40	Operation	operator: 78 operands: 49
41	ExprTuple	74
42	Operation	operator: 50 operands: 51
43	ExprTuple	86, 95
44	ExprTuple	52
45	NamedExprs	element: 53 targets: 58
46	Literal
47	ExprTuple	98
48	NamedExprs	element: 54 targets: 58
49	ExprTuple	55, 56
50	Literal
51	NamedExprs	element: 57 targets: 58
52	Literal
53	Operation	operator: 59 operands: 66
54	Operation	operator: 60 operands: 61
55	Operation	operator: 87 operands: 62
56	Operation	operator: 63 operands: 64
57	Operation	operator: 65 operands: 66
58	Operation	operator: 67 operands: 68
59	Literal
60	Literal
61	NamedExprs	operation: 69 part: 74
62	ExprTuple	95, 70
63	Literal
64	ExprTuple	71, 72
65	Literal
66	NamedExprs	state: 73 part: 74
67	Literal
68	ExprTuple	98, 75
69	Variable
70	Operation	operator: 89 operands: 76
71	Operation	operator: 91 operand: 83
72	Operation	operator: 78 operands: 79
73	Variable
74	Variable
75	Operation	operator: 80 operands: 81
76	ExprTuple	98, 82
77	ExprTuple	83
78	Literal
79	ExprTuple	84, 85
80	Literal
81	ExprTuple	86, 98
82	Operation	operator: 87 operands: 88
83	Literal
84	Operation	operator: 89 operands: 90
85	Operation	operator: 91 operand: 95
86	Variable
87	Literal
88	ExprTuple	95, 98
89	Literal
90	ExprTuple	93, 94
91	Literal
92	ExprTuple	95
93	Literal
94	Operation	operator: 96 operands: 97
95	Literal
96	Literal
97	ExprTuple	98, 99, 100, 101
98	Literal
99	Literal
100	Literal
101	Variable