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
from proveit.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import Equals, Forall, InSet
from proveit.physics.quantum import m_ket_domain, var_ket_psi
from proveit.physics.quantum.circuits import circuit__u_Akl_v, circuit__u_Akl_v__psi_m
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Lambda(var_ket_psi, Conditional(Forall(instance_param_or_params = [A11_to_Akl], instance_expr = Equals(Prob(circuit__u_Akl_v), Prob(circuit__u_Akl_v__psi_m)).with_wrapping_at(2)).with_wrapping(), InSet(var_ket_psi, m_ket_domain)))

# 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())

\lvert \psi \rangle \mapsto \left\{\begin{array}{l}\forall_{A_{1, 1}, A_{1, 2}, \ldots, A_{1, l}, A_{2, 1}, A_{2, 2}, \ldots, A_{2, l}, \ldots\ldots, A_{k, 1}, A_{k, 2}, \ldots, A_{k, l}}~\\
\left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle}
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle} \\
\qin{\lvert \psi \rangle} & { /^{m} } \qw & { /^{m} } \qw & \gate{\cdots} & { /^{m} } \qw & \qout{\lvert \psi \rangle}
} \end{array}\right) \end{array} \end{array}\right)\end{array} \textrm{ if } \lvert \psi \rangle \in \mathbb{C}^{2^{m}}\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

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

Expression of type Lambda¶

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

Expression of type Lambda ¶

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