Expression of type QcircuitEquiv ¶

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 ExprRange, Variable, VertExprArray, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _ket_u, _phase, _psi_t_ket, _s
from proveit.physics.quantum.circuits import MultiQubitElem, Output, Qcircuit, QcircuitEquiv

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Neg(t)
sub_expr3 = Add(t, one)
sub_expr4 = Add(t, _s)
sub_expr5 = Add(sub_expr2, two)
sub_expr6 = TensorProd(_psi_t_ket, _ket_u)
sub_expr7 = Interval(one, sub_expr4)
sub_expr8 = frac(one, sqrt(two))
expr = QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray([Output(state = ScalarMult(sub_expr8, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(Add(sub_expr2, one))), _phase)), ket1)))), ExprRange(sub_expr1, Output(state = ScalarMult(sub_expr8, VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))), sub_expr5, zero).with_decreasing_order(), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = Interval(sub_expr3, sub_expr4)), one, _s).with_wrapping_at(2,6)])), Qcircuit(vert_expr_array = VertExprArray([MultiQubitElem(element = Output(state = sub_expr6, part = one), targets = sub_expr7), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = sub_expr6, part = Add(sub_expr1, t)), targets = sub_expr7), sub_expr5, zero), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = sub_expr6, part = sub_expr1), targets = sub_expr7), sub_expr3, sub_expr4).with_wrapping_at(2,6)])))

# 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(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{-\left(-t + 1\right)} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
& \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
& \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 3} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
& \qout{\vdots} \\
& \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
& \qout{\lvert u \rangle}
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \multiqout{2}{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
& \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
& \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
& \qout{\vdots} \qwx[1] \\
& \qout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~0 + t~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} \qwx[1] \\
& \qout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle~\mbox{part}~t + 1~\mbox{on}~\{1~\ldotp \ldotp~t + s\}}
} \end{array}\right)

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

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

Expression of type QcircuitEquiv¶

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

Expression of type QcircuitEquiv ¶

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