Factoring with Qutrits

arXiv: 1504.03383, 1512.03824, 1605.02756
Factoring with Qutrits
Shawn Xingshan Cui
Stanford University
August 30, 2016
Joint work with Alex Bocharov, Martin Roetteler, Krysta Svore
(QuArC, Microsoft Research)
Shawn Xingshan Cui
Factoring with Qutrits
1/11
Why Qutrits?
Qubit = {|0i, |1i}
Shawn Xingshan Cui
v.s.
Qutrit = {|0i, |1i, |2i}
Factoring with Qutrits
2/11
Why Qutrits?
Qubit = {|0i, |1i}
v.s.
Qutrit = {|0i, |1i, |2i}
Multi-valued logic has computational advantage.
Shawn Xingshan Cui
Factoring with Qutrits
2/11
Why Qutrits?
Qubit = {|0i, |1i}
v.s.
Qutrit = {|0i, |1i, |2i}
Multi-valued logic has computational advantage.
Experimental implementation, e.g. linear ion traps, cold
atoms, entangled photons
Shawn Xingshan Cui
Factoring with Qutrits
2/11
Why Qutrits?
Qubit = {|0i, |1i}
v.s.
Qutrit = {|0i, |1i, |2i}
Multi-valued logic has computational advantage.
Experimental implementation, e.g. linear ion traps, cold
atoms, entangled photons
Topological quantum computation (TQC) by non-abelian
anyons. Certain anyon system naturally encodes a qutrit.
Shawn Xingshan Cui
Factoring with Qutrits
2/11
Why Qutrits?
Qubit = {|0i, |1i}
v.s.
Qutrit = {|0i, |1i, |2i}
Multi-valued logic has computational advantage.
Experimental implementation, e.g. linear ion traps, cold
atoms, entangled photons
Topological quantum computation (TQC) by non-abelian
anyons. Certain anyon system naturally encodes a qutrit.
TQC is fault tolerant; has asymptotically better efficiency.
Shawn Xingshan Cui
Factoring with Qutrits
2/11
Metaplectic Quantum Computer
Shawn Xingshan Cui
Factoring with Qutrits
3/11
Metaplectic Quantum Computer
Metaplectic Quantum Computer: SU(2)4 anyon system
(←→ fractional quantum Hall liquids at ν = 8/3)
Shawn Xingshan Cui
Factoring with Qutrits
3/11
Metaplectic Quantum Computer
Metaplectic Quantum Computer: SU(2)4 anyon system
(←→ fractional quantum Hall liquids at ν = 8/3)
Braiding and measurement
⇒ Metaplectic Basis:
Qutrit Clifford + R|2i .
X |ii = |i + 1i, Q|ii = ω δi,2 |ii;
P
H|ii = √13 2j=0 ω ij |ji;
SUM|i, ji = |i, i + ji.
R|2i = diag(1, 1, −1)
Shawn Xingshan Cui
Factoring with Qutrits
3/11
Metaplectic Basis
R|2i is non-Clifford, (not in any Clifford hierarchy).
Shawn Xingshan Cui
Factoring with Qutrits
4/11
Metaplectic Basis
R|2i is non-Clifford, (not in any Clifford hierarchy).
R|2i is obtained from the magic state ψ = |0i + |1i − |2i.
Shawn Xingshan Cui
Factoring with Qutrits
4/11
Metaplectic Basis
R|2i is non-Clifford, (not in any Clifford hierarchy).
R|2i is obtained from the magic state ψ = |0i + |1i − |2i.
ψ is produced exactly by topological measurement in 9/4 trials
on average. Much better than any state distillation method.
Shawn Xingshan Cui
Factoring with Qutrits
4/11
Metaplectic Basis
R|2i is non-Clifford, (not in any Clifford hierarchy).
R|2i is obtained from the magic state ψ = |0i + |1i − |2i.
ψ is produced exactly by topological measurement in 9/4 trials
on average. Much better than any state distillation method.
Theorem (BCKW, 2015)
Any single qutrit gate can be approximated with precision by a
metaplectic circuit with R|2i -count O(log(1/)).
(Compared with O(log3.97 (1/)) Solovay-Kitaev)
Shawn Xingshan Cui
Factoring with Qutrits
4/11
Clifford + P9 Basis
Clifford + P9 , P9 = diag(1, ω9 , ω92 ), ω9 = e
Shawn Xingshan Cui
2πi
9
.
Factoring with Qutrits
5/11
Clifford + P9 Basis
Clifford + P9 , P9 = diag(1, ω9 , ω92 ), ω9 = e
P9 :
2πi
9
.
Qutrit analog of the qubit π/8-gate.
P9 ∈ C3 3rd Clifford hierarchy.
Obtained from magic state distillation.
magic state µ = |0i + ω9 |1i + ω92 |2i.
distillation complexity: requires O(log3 (1/δ)) raw magic states
to distill one copy of µ with fidelity 1 − δ.
Shawn Xingshan Cui
Factoring with Qutrits
5/11
Clifford + P9 Basis
Clifford + P9 , P9 = diag(1, ω9 , ω92 ), ω9 = e
P9 :
2πi
9
.
Qutrit analog of the qubit π/8-gate.
P9 ∈ C3 3rd Clifford hierarchy.
Obtained from magic state distillation.
magic state µ = |0i + ω9 |1i + ω92 |2i.
distillation complexity: requires O(log3 (1/δ)) raw magic states
to distill one copy of µ with fidelity 1 − δ.
Theorem (BCRS, 2015, informal)
One can implement ternary arithmetic (e.g. ternary adder,
comparison, multiplication, subtraction, etc.) exactly over Clifford
+ P9 basis.
Shawn Xingshan Cui
Factoring with Qutrits
5/11
Emulate Qubits in Qutrit Computer
Embed a qubit {|0i, |1i} in a qutrit {|0i, |1i, |2i}.
Shawn Xingshan Cui
Factoring with Qutrits
6/11
Emulate Qubits in Qutrit Computer
Embed a qubit {|0i, |1i} in a qutrit {|0i, |1i, |2i}.
Emulate a qubit gate with
gate. E.g.,
 a qutrit
!
0 1 0
0 1


−→ 1 0 0
σX =
1 0
0 0 ∗
Shawn Xingshan Cui
Factoring with Qutrits
6/11
Emulate Qubits in Qutrit Computer
Embed a qubit {|0i, |1i} in a qutrit {|0i, |1i, |2i}.
Emulate a qubit gate with
gate. E.g.,
 a qutrit
!
0 1 0
0 1


−→ 1 0 0
σX =
1 0
0 0 ∗
Any qubit algorithm can be emulated in a qutrit computer.
Emulation efficiency?
Shawn Xingshan Cui
Factoring with Qutrits
6/11
Emulate Qubits in Qutrit Computer
Embed a qubit {|0i, |1i} in a qutrit {|0i, |1i, |2i}.
Emulate a qubit gate with
gate. E.g.,
 a qutrit
!
0 1 0
0 1


−→ 1 0 0
σX =
1 0
0 0 ∗
Any qubit algorithm can be emulated in a qutrit computer.
Emulation efficiency?
For Shor’s Algorithm, compare the cost in terms of raw magic
state count.

πi

4

Clifford + π/8(qubit) |0i + e |1i
Clifford + R|2i (qutrit) |0i + |1i − |2i


4πi
2πi
Clifford + P (qutrit)
|0i + e 9 |1i + e 9 |2i
9
Shawn Xingshan Cui
Factoring with Qutrits
6/11
Shor’s Factorization–Period Finding
Quantum part:
Given a < N, (a, N) = 1, find the smallest number r , such that
ar = 1 mod N.
1
Prepare quantum state proportional to the following
superposition:
N2
X
|ki|ak mod Ni
k=0
2
Perform quantum Fourier transform of the first register.
3
Measure the first register.
Shawn Xingshan Cui
Factoring with Qutrits
7/11
Qutrit Emulation of Shor’s Factorization
Proposition (BRS, 2016)
The cost of emulating the binary circuit of period finding is
proportional to the cost of emulating the Toffoli gate.
Shawn Xingshan Cui
Factoring with Qutrits
8/11
Qutrit Emulation of Shor’s Factorization
Proposition (BRS, 2016)
The cost of emulating the binary circuit of period finding is
proportional to the cost of emulating the Toffoli gate.
Proposition (BRS, 2016)
The Toffoli gate can be emulated exactly in the Clifford + P9 basis
either
1
by a four-qutrit circuit with 6 P9 gates (with one ancilla),
2
or by a three-qutrit circuit with 15 P9 gates (ancilla free).
Shawn Xingshan Cui
Factoring with Qutrits
8/11
Qutrit Emulation of Shor’s Factorization
Proposition (BRS, 2016)
The cost of emulating the binary circuit of period finding is
proportional to the cost of emulating the Toffoli gate.
Proposition (BRS, 2016)
The Toffoli gate can be emulated exactly in the Clifford + P9 basis
either
1
by a four-qutrit circuit with 6 P9 gates (with one ancilla),
2
or by a three-qutrit circuit with 15 P9 gates (ancilla free).
Proposition (BRS, 2016)
The P9 can be approximated by a metaplectic circuit of R|2i -count
6 log3 (1/).
Shawn Xingshan Cui
Factoring with Qutrits
8/11
Comparison of cost of implementing Toffoli
Compare the cost of Toffoli gate in qubit/qutrit models:
Clifford + π/8
Clifford A + P9
Clifford B + P9
Clifford A + R|2i
Clifford B + R|2i
Clean magic states
7
15
6
15
6
Raw resources
7(2 log2 (1/δ))2.5
15 log32 (1/δ)
6 log32 (1/δ)
90 log3 (1/δ)
36 log3 (1/δ)
Table“Clifford A ” stands for 3-qutrit emulation of the Toffoli gate and
“Clifford B ” use 4-qutrit emulation with one clean ancilla prepared with
SUM gates.
Shawn Xingshan Cui
Factoring with Qutrits
9/11
Comparison of cost of Period finding
Compare the cost of implementing period-finding circuit in
qubit/qutrit models:
Circuits
Binary QCLA
Clifford A + P9
Clifford B + P9
Clifford A + R|2i
Clifford B + R|2i
Online width
3 n − w (n) (qubits)
3 n − w (n) (qutrits)
4 n − w (n) (qutrits)
3 n − w (n) (qutrits)
4 n − w (n) (qutrits)
Offline width
7 n (6 log2 (n))2.5
15 n (3 log2 (n))3
6 n (3 log2 (n))3
90 × 3 n log3 (n)
36 × 3 n log3 (n)
Table(w (n) is the Hamming weight of n). Clifford A stands for 3-qutrit
emulation of the Toffoli gate and case B for the 4-qutrit emulation. The
last column in metaplectic rows shown the expected average of the
probabilistic width.
Shawn Xingshan Cui
Factoring with Qutrits
10/11
Summary & Conclusion
Introduced two qutrit basis:
(
Clifford + R|2i
Clifford + P9
Metaplectic TQC
Qutrit anolog of qubit Clifford + π/8
Compared the cost of Shor’s algorithm in qubit/qutrit models.
Shawn Xingshan Cui
Factoring with Qutrits
11/11
Summary & Conclusion
Introduced two qutrit basis:
(
Clifford + R|2i
Clifford + P9
Metaplectic TQC
Qutrit anolog of qubit Clifford + π/8
Compared the cost of Shor’s algorithm in qubit/qutrit models.
⇒ The solutions over the metaplectic architecture are the most
cost-effective in both asymptotic and practical sense. Plus bonus:
naturally fault-tolerate.
Shawn Xingshan Cui
Factoring with Qutrits
11/11
Summary & Conclusion
Introduced two qutrit basis:
(
Clifford + R|2i
Clifford + P9
Metaplectic TQC
Qutrit anolog of qubit Clifford + π/8
Compared the cost of Shor’s algorithm in qubit/qutrit models.
⇒ The solutions over the metaplectic architecture are the most
cost-effective in both asymptotic and practical sense. Plus bonus:
naturally fault-tolerate.
Thank you!
Shawn Xingshan Cui
Factoring with Qutrits
11/11

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