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Expression of type Implies

from the theory of proveit.physics.quantum.QPE

In [1]:
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, m, n
from proveit.logic import Implies, NotEquals
from proveit.numbers import Add, Interval, one
from proveit.physics.quantum import NumKet, Z, ket_plus
from proveit.physics.quantum.QPE import QPE, _U, _ket_u, _phase_est_circuit, _s, _s_wire, _t
from proveit.physics.quantum.circuits import Gate, Input, Measure, MultiQubitElem, Output, Qcircuit
In [2]:
# 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(one, _t)
sub_expr5 = Interval(sub_expr2, sub_expr3)
sub_expr6 = [ExprRange(sub_expr1, Measure(basis = Z), one, _t), _s_wire]
sub_expr7 = MultiQubitElem(element = Gate(operation = QPE(_U, _t), part = sub_expr1), targets = Interval(one, sub_expr3))
sub_expr8 = ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)
sub_expr9 = [ExprRange(sub_expr1, Input(state = ket_plus), one, _t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr5), one, _s)]
sub_expr10 = [ExprRange(sub_expr1, sub_expr7, one, _t), ExprRange(sub_expr1, sub_expr7, sub_expr2, sub_expr3)]
sub_expr11 = VertExprArray(sub_expr9, sub_expr10, sub_expr6, [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(n, _t), part = sub_expr1), targets = sub_expr4), one, _t), sub_expr8])
expr = Implies(NotEquals(VertExprArray(sub_expr9, sub_expr10, sub_expr6, [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(m, _t), part = sub_expr1), targets = sub_expr4), one, _t), sub_expr8]), sub_expr11), NotEquals(_phase_est_circuit, Qcircuit(vert_expr_array = sub_expr11))).with_wrapping_at(2)
expr:
In [3]:
# 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
In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\begin{array}{c} \begin{array}{l} \left(\left(\begin{array}{cccc} 
 \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert + \rangle} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \meter 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert m \rangle_{t}~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array} \\
\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert + \rangle} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \meter 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert m \rangle_{t}~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array} \\
\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array} & \vdots & \begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array} & \vdots \\
\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert + \rangle} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \meter 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert m \rangle_{t}~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array} \\
\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~t + 1~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array} \\
\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~t + 2~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array} \\
\vdots & \vdots & \begin{array}{c}:\\ \left(s - 3\right) \times \\:\end{array} & \vdots \\
\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~t + s~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array} \\
\end{array}
\right) \neq \left(\begin{array}{cccc} 
 \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert + \rangle} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \meter 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert n \rangle_{t}~\mbox{part}~1~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array} \\
\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert + \rangle} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \meter 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert n \rangle_{t}~\mbox{part}~2~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array} \\
\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array} & \vdots & \begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array} & \vdots \\
\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert + \rangle} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \meter 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert n \rangle_{t}~\mbox{part}~t~\mbox{on}~\{1~\ldotp \ldotp~t\}} 
} \end{array} \\
\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~t + 1~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array} \\
\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~t + 2~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array} \\
\vdots & \vdots & \begin{array}{c}:\\ \left(s - 3\right) \times \\:\end{array} & \vdots \\
\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \qin{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{\textrm{QPE}\left(U, t\right)~\mbox{part}~t + s~\mbox{on}~\{1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array} & \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qout{\lvert u \rangle~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} 
} \end{array} \\
\end{array}
\right)\right) \Rightarrow  \\ \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert m \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m \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 m \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert m \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) \neq \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert n \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert n \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 n \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert n \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right)\right) \end{array} \end{array}
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.()(2)('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 6
operands: 5
4Operationoperator: 6
operands: 7
5ExprTuple10, 12
6Literal
7ExprTuple8, 9
8Operationoperator: 11
operands: 10
9Operationoperator: 11
operands: 12
10ExprTuple14, 15, 16, 13
11Literal
12ExprTuple14, 15, 16, 17
13ExprTuple18, 26
14ExprTuple19, 20
15ExprTuple21, 22
16ExprTuple23, 24
17ExprTuple25, 26
18ExprRangelambda_map: 27
start_index: 97
end_index: 98
19ExprRangelambda_map: 28
start_index: 97
end_index: 98
20ExprRangelambda_map: 29
start_index: 97
end_index: 99
21ExprRangelambda_map: 30
start_index: 97
end_index: 98
22ExprRangelambda_map: 30
start_index: 84
end_index: 85
23ExprRangelambda_map: 31
start_index: 97
end_index: 98
24ExprRangelambda_map: 32
start_index: 97
end_index: 99
25ExprRangelambda_map: 33
start_index: 97
end_index: 98
26ExprRangelambda_map: 34
start_index: 97
end_index: 99
27Lambdaparameter: 83
body: 35
28Lambdaparameter: 83
body: 36
29Lambdaparameter: 83
body: 37
30Lambdaparameter: 83
body: 38
31Lambdaparameter: 83
body: 39
32Lambdaparameter: 83
body: 40
33Lambdaparameter: 83
body: 41
34Lambdaparameter: 83
body: 43
35Operationoperator: 52
operands: 44
36Operationoperator: 68
operands: 45
37Operationoperator: 52
operands: 46
38Operationoperator: 52
operands: 47
39Operationoperator: 48
operands: 49
40Operationoperator: 69
operands: 50
41Operationoperator: 52
operands: 51
42ExprTuple83
43Operationoperator: 52
operands: 53
44NamedExprselement: 54
targets: 62
45NamedExprsstate: 55
46NamedExprselement: 56
targets: 64
47NamedExprselement: 57
targets: 58
48Literal
49NamedExprsbasis: 59
50NamedExprsoperation: 60
51NamedExprselement: 61
targets: 62
52Literal
53NamedExprselement: 63
targets: 64
54Operationoperator: 74
operands: 65
55Operationoperator: 66
operand: 79
56Operationoperator: 68
operands: 75
57Operationoperator: 69
operands: 70
58Operationoperator: 76
operands: 71
59Literal
60Literal
61Operationoperator: 74
operands: 72
62Operationoperator: 76
operands: 73
63Operationoperator: 74
operands: 75
64Operationoperator: 76
operands: 77
65NamedExprsstate: 78
part: 83
66Literal
67ExprTuple79
68Literal
69Literal
70NamedExprsoperation: 80
part: 83
71ExprTuple97, 85
72NamedExprsstate: 81
part: 83
73ExprTuple97, 98
74Literal
75NamedExprsstate: 82
part: 83
76Literal
77ExprTuple84, 85
78Operationoperator: 89
operands: 86
79Literal
80Operationoperator: 87
operands: 88
81Operationoperator: 89
operands: 90
82Literal
83Variable
84Operationoperator: 92
operands: 91
85Operationoperator: 92
operands: 93
86ExprTuple94, 98
87Literal
88ExprTuple95, 98
89Literal
90ExprTuple96, 98
91ExprTuple98, 97
92Literal
93ExprTuple98, 99
94Variable
95Literal
96Variable
97Literal
98Literal
99Literal