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 import U_1_to_i, V_1_to_j
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import one
from proveit.physics.quantum import m_ket_domain, var_ket_psi
from proveit.physics.quantum.circuits import circuit_Ui_Vj, circuit_Ui_psi_m, circuit_psi_m_Vj
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Lambda(var_ket_psi, Conditional(Forall(instance_param_or_params = [U_1_to_i, V_1_to_j], instance_expr = Implies(Equals(Prob(circuit_Ui_psi_m), one), Equals(Prob(circuit_Ui_Vj), Prob(circuit_psi_m_Vj)).with_wrapping_at(2)).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_{U_{1}, U_{2}, \ldots, U_{i}, V_{1}, V_{2}, \ldots, V_{j}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\U_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{i} \\ \downarrow\end{array}} & \qout{\lvert \psi \rangle}
} \end{array}\right) = 1\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\U_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{i} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{j} \\ \downarrow\end{array}} & \qw
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert \psi \rangle} & \gate{\begin{array}{c} \uparrow \\V_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{j} \\ \downarrow\end{array}} & { /^{m} } \qw
} \end{array}\right) \end{array} \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: 71 body: 2
1	ExprTuple	71
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	71, 10
9	Lambda	parameters: 11 body: 12
10	Operation	operator: 13 operands: 14
11	ExprTuple	42, 44
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	17, 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	Operation	operator: 21 operands: 22
19	Operation	operator: 24 operands: 23
20	Operation	operator: 24 operands: 25
21	Literal
22	ExprTuple	26, 74
23	ExprTuple	27, 73
24	Literal
25	ExprTuple	28, 29
26	Literal
27	Operation	operator: 32 operand: 34
28	Operation	operator: 32 operand: 35
29	Operation	operator: 32 operand: 36
30	ExprTuple	34
31	ExprTuple	35
32	Literal
33	ExprTuple	36
34	Operation	operator: 39 operands: 37
35	Operation	operator: 39 operands: 38
36	Operation	operator: 39 operands: 40
37	ExprTuple	42, 41
38	ExprTuple	42, 44
39	Literal
40	ExprTuple	43, 44
41	ExprTuple	45
42	ExprRange	lambda_map: 46 start_index: 73 end_index: 47
43	ExprTuple	48
44	ExprRange	lambda_map: 49 start_index: 73 end_index: 50
45	ExprRange	lambda_map: 51 start_index: 73 end_index: 74
46	Lambda	parameter: 72 body: 52
47	Variable
48	ExprRange	lambda_map: 53 start_index: 73 end_index: 74
49	Lambda	parameter: 72 body: 54
50	Variable
51	Lambda	parameter: 72 body: 55
52	IndexedVar	variable: 56 index: 72
53	Lambda	parameter: 72 body: 57
54	IndexedVar	variable: 58 index: 72
55	Operation	operator: 61 operands: 60
56	Variable
57	Operation	operator: 61 operands: 62
58	Variable
59	ExprTuple	72
60	NamedExprs	element: 63 targets: 65
61	Literal
62	NamedExprs	element: 64 targets: 65
63	Operation	operator: 66 operands: 68
64	Operation	operator: 67 operands: 68
65	Operation	operator: 69 operands: 70
66	Literal
67	Literal
68	NamedExprs	state: 71 part: 72
69	Literal
70	ExprTuple	73, 74
71	Variable
72	Variable
73	Literal
74	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 ¶