A cute robot building itself with artificial intelligence,
pencil
drawing - DALL·E 2
A cute robot building itself with artificial intelligence,
pencil
drawing - DALL·E 2
A cute robot building itself with artificial intelligence,
pencil
drawing - DALL·E 2
Hierarchical Quantum Circuit Representations
A cute robot building itself with artificial intelligence,
pencil
drawing - DALL·E 3
Outline
HierarQcal
Hierarchical Quantum Circuit
Representations
Design and implement hierarchical compute
graphs
Theoretical framework behind package
This talk
- Background
- Overview of Representation
- Results and usage
Background
Neural Networks
AlexNet
Automation
Manual
$\rightarrow$
Feature Engineering
Automated
Automation
Manual
$\rightarrow$
Architecture Engineering?
Automated
Neural Architecture Search
$^*$Fig: Abstract illustration of Neural Architecture Search methods. A
search
strategy selects an architecture A from a predefined search space A. The
architecture is passed to a performance estimation strategy, which returns the
estimated performance of A to the search strategy.
Neural Architecture Search
$^*$Fig: An illustration of different architecture spaces. Each node
in the graphs corresponds to a layer in a neural network, e.g., a convolutional or
pooling layer. Different layer types are visualized by different colors. An edge
from layer $L_i$ to layer $L_j$ denotes that $L_j$ receives the output of $L_i$ as
input. Left: an element of a chain-structured space. Right: an element of a more
complex search space with additional layer types and multiple branches and skip
connections.
Neural Architecture Search
$^*$Fig: An example of a three-level hierarchical architecture
representation. The bottom row shows how level-1 primitive operations $o^{(1)}_1$ ,
$o^{(1)}_2$ , $o^{(1)}_3$ are assembled into a level-2 motif $o^{(2)}_1$ . The top
row shows how level-2 motifs $o^{(2)}_1$ , $o^{(2)}_2$ , $o^{(2)}_3$ are then
assembled into a level-3 motif $o^{(3)}_1$.
Ansatz Design
$^*$Fig: The concept of QCNNs. $a$, Simplified illustration of classical
CNNs. A sequence of image-processing layers transforms an input image into a series
of feature maps (blue rectangles) and finally into an output probability
distribution (purple bars). C, convolution; P, pooling; FC, fully connected. $b$,
QCNNs inherit a similar layered structure. Boxes represent unitary gates or
measurement with feed-forwarding. c, The QCNN and the MerA share the same circuit
structure, but run in reverse directions.
Ansatz Design
$^*$Fig: TTN and MERA classifiers for eight qubits. The quantum circuit is
illustrated by the regions outlined in solid lines comprising inputs $\psi$, unitary
blocks $\{U_i\}_{i=1}^7$ and $\{D_i\}_{i=1}^4$, and a measurement operator M. The
dashed lines represent its conjugate transpose. The solid and dashed regions
together describe a tensor network operating on input $\psi_{1-8}$ and evaluating to
the expectation value of observable $M$
Representation
Motifs
Motifs
Motifs
Motifs
Motifs
Motifs
Motifs
Motifs
Motifs
Motifs
Motifs
Examples
hierq = Qinit(5)
hierq = Qinit(5) + Qcycle(stride=1, step=1, offset=0)
hierq = Qinit(8) + Qcycle(stride=1, step=1, offset=0)
Examples
hierq = Qinit(5)
hierq = Qinit(5) + Qcycle(stride=1, step=1, offset=0)
hierq = Qinit(8) + Qcycle(stride=1, step=1, offset=0)
Examples
hierq = Qinit(5) + Qcycle(stride=1, step=1, offset=0)
hierq = Qinit(8) + Qcycle(stride=1, step=1, offset=0)
hierq = Qinit(8) + Qcycle(stride=1, step=1, offset=0, boundary="open")
Examples
hierq = Qinit(8) + Qcycle(stride=1, step=1, offset=0)
hierq = Qinit(8) + Qcycle(stride=1, step=1, offset=0, boundary="open")
hierq = Qinit(8) + Qcycle(stride=2, step=1, offset=0, boundary="open")
Examples
hierq = Qinit(8) + Qcycle(stride=1, step=1, offset=0, boundary="open")
hierq = Qinit(8) + Qcycle(stride=2, step=1, offset=0, boundary="open")
hierq = Qinit(8) + Qcycle(stride=3, step=1, offset=0, boundary="open")
Examples
hierq = Qinit(8) + Qcycle(stride=2, step=1, offset=0, boundary="open")
hierq = Qinit(8) + Qcycle(stride=3, step=1, offset=0, boundary="open")
hierq = Qinit(8) + Qcycle(stride=1, step=2, offset=0, boundary="open")
Examples
hierq = Qinit(8) + Qcycle(stride=3, step=1, offset=0, boundary="open")
hierq = Qinit(8) + Qcycle(stride=1, step=2, offset=0, boundary="open")
hierq = Qinit(8) + Qcycle(stride=1, step=3, offset=0, boundary="open")
Examples
hierq = Qinit(8) + Qcycle(stride=1, step=2, offset=0, boundary="open")
hierq = Qinit(8) + Qcycle(stride=1, step=3, offset=0, boundary="open")
hierq = Qinit(8) + Qcycle(stride=1, step=3, offset=1, boundary="open")
Examples
hierq = Qinit(8) + Qcycle(stride=1, step=3, offset=0, boundary="open")
hierq = Qinit(8) + Qcycle(stride=1, step=3, offset=1, boundary="open")
hierq = Qinit(8) + Qcycle(1,1,0, mapping = u2)
Examples
hierq = Qinit(8) + Qcycle(1,1,0, mapping = u2)
hierq(backend="qiskit")
hierq(backend="pennylane")
Examples
hierq = Qinit(8) + Qcycle(1,1,0, mapping = u2)
hierq(backend="qiskit")
hierq(backend="pennylane")
Examples
hierq = Qinit(8) + Qmask("10100000")
hierq = Qinit(8) + Qmask("1*")
hierq = Qinit(8) + Qmask("!00")
Examples
hierq = Qinit(8) + Qmask("10100000")
hierq = Qinit(8) + Qmask("1*")
hierq = Qinit(8) + Qmask("!00")
Examples
hierq = Qinit(8) + Qmask("1*")
hierq = Qinit(8) + Qmask("!00")
hierq = Qinit(8) + Qmask("!*")
Examples
hierq = Qinit(8) + Qmask("!00")
hierq = Qinit(8) + Qmask("!*")
hierq = Qinit(8) + Qcycle(mapping = u2) + Qmask("*!") + Qcycle(mapping = u2)
Examples
hierq = Qinit(8) + Qcycle(mapping = u2) + Qmask("*!") + Qcycle(mapping = u2)
Examples
cycle = Qcycle(mapping = u2)
hierq = Qinit(8) + cycle + Qmask("*!", mapping=v2) + cycle
Examples
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("*!", mapping=v2)
hierq = Qinit(8) + cycle_mask
hierq = Qinit(8) + cycle_mask * 3
Examples
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("*!", mapping=v2)
hierq = Qinit(8) + cycle_mask
hierq = Qinit(8) + cycle_mask * 3
Examples
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("*!", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("!*", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("!*!", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("*!*", mapping=v2)
hierq = Qinit(8) + cycle_mask * 3
Examples
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("*!", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("!*", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("!*!", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("*!*", mapping=v2)
hierq = Qinit(8) + cycle_mask * 3
Examples
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("*!", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("!*", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("!*!", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("*!*", mapping=v2)
hierq = Qinit(8) + cycle_mask * 3
Examples
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("*!", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("!*", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("!*!", mapping=v2)
cycle_mask = Qcycle(1, 1, 0, mapping=u2) + Qmask("*!*", mapping=v2)
hierq = Qinit(8) + cycle_mask * 3
Results and Usage
Music Genre Classification
Random Search
# Example Search Space
strides = range(1, 8, 1)
steps = range(1, 8, 1)
mappings = [a, b, c, d, e, f, g, h]
boundaries = ["open", "periodic"]
patterns = ["10", "01", "*!", "!*", "!*!", "*!*"]
qcnn = Qinit(8) + (Qcycle(...) + Qmask(...)) * 3
Music Genre Classification
$^*$Reference Architecture:
Ansatz:
Evolutionary Search
Evolutionary Search
Tournament Selection
Tournament Selection
Mutation
Crossover
Repeat
Quantum Phase Recognition
Cluster-Ising Hamiltonian:
\[
\begin{align}
H & = -J\sum_{i=1}^{N-2} Z_iX_{i+1}Z_{i+2} - h_1\sum_{i=1}^{N} X_i - h_2\sum_{i=1}^{N-1}
X_iX_{i+1}.\nonumber
\end{align}
\]
- Given an unknown ground state $\ket{\psi_g}$, what is the phase?
- Paramagnetic
- $\mathbb{Z_2}\times\mathbb{Z_2}$ Symmetry-Protected Topological (SPT)
- Antiferromagnetic
Quantum Phase Recognition
u0 = Qinit(4) + Qpermute(share_weights=False, mapping=Ugm)
u1 = Qcycle(1,3,2, mapping=u0, boundary="open", edge_order=[1,3,2,4])
u2 = sum([Qcycle(1,3,offset, mapping=Ugm, boundary="open") for offset in range(3)])
u3 = Qmask("101",1,3,0, mapping=Vgm, boundary="open")
u4 = Qcycle(1, mapping=Ugm, boundary="open")
motif = Qinit(15) + u1 + u2 + u3 + u4
Quantum Phase Recognition
Reference1
$\rightarrow$
Found2
Quantum Phase Recognition
Found:
What's next?
Compute Graph Design
- Quantum Circuits
- Classical Circuits
- Tensor Networks
- Neural Networks
Exponentiation modulo $N$
Addition
def or(bits, symbols=None, state=None):
b1, b2 = state[bits[0]], state[bits[1]]
state[bits[0]] = b1 or b2
return state
tensors = [np.array([1, 0])] * 5
hierq = (Qinit(5, tensors=tensors)
+ mask_anc
+ U_psi + grover)
psi = hierq()
?
Paper
Hierarqcal
Slides
Programming
Shor's Algorithm
- Modular Exponentiation
- Quantum Fourier Transform
Shor's Algorithm
- Modular Exponentiation
- Ctrl-Mult modulo $N$
- Quantum Fourier Transform
Shor's Algorithm
- Modular Exponentiation
- Ctrl-Mult modulo $N$
- Addition modulo $N$
- Ctrl-Mult modulo $N$
- Quantum Fourier Transform
Shor's Algorithm
- Modular Exponentiation
- Ctrl-Mult modulo $N$
- Addition modulo $N$
- Addition
- Addition modulo $N$
- Ctrl-Mult modulo $N$
- Quantum Fourier Transform
Shor's Algorithm
- Modular Exponentiation
- Ctrl-Mult modulo $N$
- Addition modulo $N$
- Addition
- Carry
- xor
- Addition
- Addition modulo $N$
- Ctrl-Mult modulo $N$
- Quantum Fourier Transform
Shor's Algorithm
- Modular Exponentiation
- Ctrl-Mult modulo $N$
- Addition modulo $N$
- Addition
- Carry
- cnot
- Toffoli
- xor
- cnot
- Carry
- Addition
- Addition modulo $N$
- Ctrl-Mult modulo $N$
- Quantum Fourier Transform
Shor circuit
# ====== Motifs level 1
carry_motif = (
Qinit(4)
+ Qmotif(E=[(1, 2, 3)], mapping=toffoli)
+ Qmotif(E=[(1, 2)], mapping=cnot)
+ Qmotif(E=[(0, 2, 3)], mapping=toffoli)
)
# Turn carry into a function to be able to reverse it easily
carry = lambda r: carry_motif if r == 1 else carry_motif.reverse()
plot_circuit(carry(1))
plot_circuit(carry(-1))
Shor circuit
# ====== Motifs level 1
carry_motif = (
Qinit(4)
+ Qmotif(E=[(1, 2, 3)], mapping=toffoli)
+ Qmotif(E=[(1, 2)], mapping=cnot)
+ Qmotif(E=[(0, 2, 3)], mapping=toffoli)
)
# Turn carry into a function to be able to reverse it easily
carry = lambda r: carry_motif if r == 1 else carry_motif.reverse()
plot_circuit(carry(1))
plot_circuit(carry(-1))
Shor circuit
# ====== Motifs level 1
sum = lambda r=1: Qinit(3, name=f"sum") + Qpivot(
"*1", merge_within="01", mapping=cnot, edge_order=[-1 * r]
)
plot_circuit(sum(1))
plot_circuit(sum(-1))
Shor circuit
# ====== Motifs level 1
sum = lambda r=1: Qinit(3, name=f"sum") + Qpivot(
"*1", merge_within="01", mapping=cnot, edge_order=[-1 * r]
)
plot_circuit(sum(1))
plot_circuit(sum(-1))
Shor circuit
# ====== Motifs level 2
carry_sum_motif = (
lambda r=1: Qinit(4, name=f"crs")
+ Qpivot("1*", merge_within="1111", mapping=carry(-r))
+ Qpivot("1*", merge_within="111", mapping=sum(r))
)
carry_sum = lambda r=1: carry_sum_motif(1) if r == 1 else carry_sum_motif(-1).reverse()
plot_circuit(carry_sum(1))
plot_circuit(carry_sum(-1))
Shor circuit
# ====== Motifs level 2
carry_sum_motif = (
lambda r=1: Qinit(4, name=f"crs")
+ Qpivot("1*", merge_within="1111", mapping=carry(-r))
+ Qpivot("1*", merge_within="111", mapping=sum(r))
)
carry_sum = lambda r=1: carry_sum_motif(1) if r == 1 else carry_sum_motif(-1).reverse()
plot_circuit(carry_sum(1))
plot_circuit(carry_sum(-1))
Shor circuit
cnot_sum_motif = (
lambda r=1: Qinit(3, name=f"cns")
+ Qpivot("*1", merge_within="11", mapping=cnot)
+ Qpivot("*1", merge_within="111", mapping=sum(r))
)
cnot_sum = lambda r=1: cnot_sum_motif(1) if r == 1 else cnot_sum_motif(-1).reverse()
plot_circuit(cnot_sum(1))
plot_circuit(cnot_sum(-1))
Shor circuit
cnot_sum_motif = (
lambda r=1: Qinit(3, name=f"cns")
+ Qpivot("*1", merge_within="11", mapping=cnot)
+ Qpivot("*1", merge_within="111", mapping=sum(r))
)
cnot_sum = lambda r=1: cnot_sum_motif(1) if r == 1 else cnot_sum_motif(-1).reverse()
plot_circuit(cnot_sum(1))
plot_circuit(cnot_sum(-1))
Shor circuit
# ====== Motifs level 3
carry_layer = lambda r=1: Qcycle(
1,
3,
0,
mapping=carry(r),
boundary="open",
edge_order=[r],
)
plot_circuit(carry_layer_1)
Shor circuit
cnot_sum_pivot = lambda r: Qpivot("*10", merge_within="111", mapping=cnot_sum(r))
plot_circuit(Qinit(6) + cnot_sum_pivot(1))
plot_circuit(Qinit(8) + cnot_sum_pivot(1))
Shor circuit
cnot_sum_pivot = lambda r: Qpivot("*10", merge_within="111", mapping=cnot_sum(r))
plot_circuit(Qinit(6) + cnot_sum_pivot(1))
plot_circuit(Qinit(8) + cnot_sum_pivot(1))
Shor circuit
carry_sum_layer = lambda r: (
Qmask("*111")
+ Qcycle(1, 3, 0,
boundary="open",
mapping=carry_sum(r),
edge_order=[-r],
)
+ Qunmask("previous")
)
plot_circuit(Qinit(10) + carry_sum_layer(1))
Shor circuit
addition = carry_layer(1) + cnot_sum_pivot(1) + carry_sum_layer(1)
subtraction = carry_sum_layer(-1) + cnot_sum_pivot(-1) + carry_layer(-1)
plot_circuit(Qinit(11) + addition)
plot_circuit(Qinit(11) + subtraction)
plot_circuit(Qinit(13) + addition)
plot_circuit(Qinit(13) + subtraction)
Shor circuit
addition = carry_layer(1) + cnot_sum_pivot(1) + carry_sum_layer(1)
subtraction = carry_sum_layer(-1) + cnot_sum_pivot(-1) + carry_layer(-1)
plot_circuit(Qinit(11) + addition)
plot_circuit(Qinit(11) + subtraction)
plot_circuit(Qinit(13) + addition)
plot_circuit(Qinit(13) + subtraction)
Shor circuit
addition = carry_layer(1) + cnot_sum_pivot(1) + carry_sum_layer(1)
subtraction = carry_sum_layer(-1) + cnot_sum_pivot(-1) + carry_layer(-1)
plot_circuit(Qinit(11) + addition)
plot_circuit(Qinit(11) + subtraction)
plot_circuit(Qinit(13) + addition)
plot_circuit(Qinit(13) + subtraction)
Shor circuit
addition = carry_layer(1) + cnot_sum_pivot(1) + carry_sum_layer(1)
subtraction = carry_sum_layer(-1) + cnot_sum_pivot(-1) + carry_layer(-1)
plot_circuit(Qinit(11) + addition)
plot_circuit(Qinit(11) + subtraction)
plot_circuit(Qinit(13) + addition)
plot_circuit(Qinit(13) + subtraction)
Shor circuit
exp_mod_n = tuple()
for k in range(n):
exp_mod_n += (
Qpivot(mapping=ctrl_mult(a ** (2**k), N, n, ctrl=k, divide=False))
+ swap_bx
+ Qpivot(mapping=ctrl_mult(a ** (2**k), N, n, ctrl=k, divide=True))
)
hierq = Qinit(nq, tensors=tensors) + exp_mod_n
Quantum Fourier Transform
# QFT circuit
n = 4
qft = (
Qpivot(mapping=Qunitary("h()^0"))
+ Qpivot(
mapping=Qunitary("cp(x)^01"),
share_weights=False,
symbol_fn=lambda x, ns, ne: np.pi * 2 ** (-ne),
)
+ Qmask("1*")
) * n
qft_n = Qinit(n) + qft
circuit_qft = qft_n(backend="qiskit", barriers=False)
circuit_qft.draw("mpl")
Classical Adder
def half_adder(bits, symbols=None, state=None):
b1, b2 = state[bits[0]], state[bits[1]]
xor = b1 ^ b2
carry = b1 and b2
state[bits[0]] = carry
state[bits[1]] = xor
return state
def or_top(bits, symbols=None, state=None):
b1, b2 = state[bits[0]], state[bits[1]]
state[bits[0]] = b1 or b2
return state
Classical Adder
def half_adder(bits, symbols=None, state=None):
b1, b2 = state[bits[0]], state[bits[1]]
xor = b1 ^ b2
carry = b1 and b2
state[bits[0]] = carry
state[bits[1]] = xor
return state
def or_top(bits, symbols=None, state=None):
b1, b2 = state[bits[0]], state[bits[1]]
state[bits[0]] = b1 or b2
return state
Classical Adder
# program
full_adder = (
Qinit(3)
+ Qcycle(mapping=half_adder, boundary="open")
+ Qpivot(global_pattern="1*", merge_within="11"
,mapping=or_top)
)
addition = (
Qinit(n)
+ Qpivot("*1", "11", mapping=half_adder)
+ Qcycle(step=2,
edge_order=[-1],
mapping=full_adder,
boundary="open")
)