Transcript Inverting Well Conditioned Matrices in Quantum LogSpace Amnon Ta-Shma Tel-Aviv University Space Bounded Complexity Space complexity measures the memory size needed for solving a problem.

Inverting Well Conditioned
Matrices in Quantum
LogSpace
Amnon Ta-Shma
Tel-Aviv University
Space Bounded Complexity
Space complexity measures the memory
size needed for solving a problem.
An Example: Multiplying two
matrices
Input:
Two n
n matrices A,B.
Output:
C = AB.
Algorithm: Ci,j = Ai,k Bk,j
For i=1,..,n
For j=1,..,n
{
c=0;
For k=1,..,n
c = c + Ai,k Bk,j;
output c;
}
We do not count the input as
working area, because we are
notWe
allowed
change
do nottocount
the it.
output as
working area, because we are not
We only count memory
allowed to read or change it. We
elements that we can
view it as sending the output to a
read, write and change
printer.
(i,j,c,k) as working area.
An Example: Multiplying two
matrices
Input:
Two n
n matrices A,B.
Output:
C = AB.
Algorithm: Ci,j = Ai,k Bk,j
• The input is not counted
• The output is not counted
• The only thing we count is memory we can read,
write and change.
The algorithm above runs in O(log n) space.
Another Example: Undirected
connectivity
Input: An undirected graph G=(V,E).
Output: Is the graph connected?
• Can be solved with linear space and time.
• Omer Reingold showed the problem can be solved
with logarithmic space and polynomial time.
Problems not known to be in Log
NL – complete.
• Connectivity of directed graphs.
• Determinant of an integer matrix.
• Inverting an integer matrix.
DET – complete.
NL – Non-deterministic Logspace.
DET – all languages that are LogSpace
reducible to integer determinant.
What is known
Log
NL
DET
STCON is NL
complete
DSPACE(log2n)
Matrix inversion, int
determinant are DET
complete.
Probabilistic space-bounded
computation
BPL – the class of languages that are
solvable by space-bounded machines that
have online access to an unbounded
sequence of truly uniform bits.
Log
BPL
BPL DET
DSPACE(log1.5n)
[SaksZhou]
Quantum space-bounded
computation
BQL – all languages solvable by a LOG
machine that may use O(log n) qubits.
• Counting the number of qubits is a natural
complexity measure.
• The definition has several variants, and we
will discuss it soon.
What do we know about BQL?
Log
BPL
BQL
DSPACE(log2n)
Not much else is known.
No natural candidate for a language in BQL
not known to be in BPL.
In this talk
• We will modify an algorithm of Harrow,
Hassidim, Lloyd for approximated matrix
inversion.
• HHL studied quantum time complexity.
We will study quantum space cpmplexity.
• We will show the problem is in BQL
• The problem is not known to be in BPL.
Quadratic gap
• This is first natural candidate for a problem
in BQL not in BPL.
• It Presents a quadratic gap between BQL
and what we currently know in BPL, and
this gap is best possible.
• Our work might lead to new classical
algorithms.
Defining BQL
Deterministic Space TM
• Input tape: Read only, Head moves in all directions
• Output tape: Write only, Head moves Left
• Work tape: Read/Write, All directions.
Quantum space-bounded
machines
• An additional quantum tape with O(log n) qubits.
• Two heads over the quantum tape.
• The allowed quantum operations are:
HAD, CNOT, T plus
Intermediate
measurement
measurements M in the standard basis.
H T
M
X
T
H
H
T
X
T
X
T
X
M
H
X
T
X M
Classical control
We use the usual function mechanism. The
function only depends on the classical data.
:Qx
Input x
Work
Qx
Work x
4
x
{L,R}
out
 (qCNOT) applies CNOT on the qubits under the two heads
 Similarly for (qHAD) and (qT)
 (qM) measures the qubit under the first head in the
standard basis. Moves to qM,0,qM,1 depending on answer.
BQL
• O(log n) classical bits and qubits.
• Classical control.
• Intermediate measurements.
H T
M
X
T
H
H
T
X
T
X
T
X
M
H
X
T
X M
BQL without intermediate measurements is also
interesting but possibly much weaker.
Matrix inversion and the HHL
algorithm
Time complexity of Matrix
inversion
• Can be solved as fast as matrix
multiplication. Current best time O(n ),
2.37.
• Matrix inversion depends on all input bits
and so the time complexity must be (n2).
The HHL problem
The HHL algorithm studies a modified
version of matrix inversion:
Input: A matrix A, a vector b,
Output: Approximation of certain predicates
of x=A-1b
Since we deal with approximation, the input
matrix has to be stable.
Stability – the condition number
• Matrix inversion is not stable, if there
exists an eigenvalue close to 0.
• Matrix inversion is stable if all eigenvalues
are far from zero.
The condition number (A) is defined to be
(A)= ||A|| / ||A-1||
HHL’09
Input:
A matrix A, a vector b,
condition number k
Output: Approximation of certain
predicates of x=A-1b
• Quantum Time complexity: O(k log n).
• Exponentially faster than the classical time
bound Ω(n).
HHL - Summary
•
•
•
•
•
Only an approximation
Only for well conditioned A
Only for sparse matrices A
Only for special b
Only for certain predicates over x.
Very nice idea!
Surprising technique and result.
Our result
Input:
A matrix A,
condition number k
Output: Approximation of A-1
• Quantum space complexity: O(log(kn)).
• Currently best classical bound O(log2n).
Quadratic gap.
The technique
Basic idea:
Sampling the spectrum
using phase estimation
First observation: We can work
with Hermitian matrices
Given input A. We look for the SVD, A=UDV.
Define H=
0 A
, H is Hermitian.
A† 0
The SVD of H is
U 0
0 D
U† 0
0 V†
D† 0
0 V
And it so happens that one can read A’s
decomposition from H’s decomposition.
We also assume all eigenalues are well-separated.
Basic approach
Input: Hermitian A.
U=eiA is unitary.
Assume:
• We can simulate Ut for t=1,…,T, and,
• We know an eigenvector v of U.
Then, using phase estimation, we estimate
the eigenvalue λ associated with v.
First challenge:
How do we find an eigenvector?
Classically: A big question.
Once we know A and an eigenvector v,
We can easily compute in small space.
Sampling instead of finding
an eigenvector
The completely mixed state I is the mixture
Obtained by taking a uniform eigenvector of A.
If we apply phase estimation on I we sample
a random (eigenvector,eigenvalue) pair of A.
We can generate the uniform distribution over
the eigenvectors of A, even though we do not
know any specific eigenvecctor.
2nd Challenge: Simulate U=eiA
{HAD, CNOT, T} is a universal basis,
Hence any unitary U can be approximated
by a circuit with these gates.
The challenge is designing a deterministic
Log space algorithm that given A produces
a quantum circuit over {HAD, CNOT, T}
That approximates U=eiA.
Reminder:
A unitary that acts non-trivially
only on a 2 dimensional
Universality
of {HAD, CNOT, T}
Given a
subspace spanned by 2
standardU:
basis vectors.
unitary
1. Decompose U to a product of 2-level
unitaries.
Using the Solovay
Theorem.
2. Convert a 2-level unitary to Kitaev
a product
of
CNOT and 1-qubit unitaries.
3. Approximate any 1-qubit unitary by a short
product of {HAD, T}
Simulating U=eiA in small space
Given a unitary U:
1. Approximately decompose it to a product
of 2-level unitaries, using Trotter formula.
2. Convert a 2-level unitary to a product of
CNOT and 1-qubit unitaries.
3. Approximate any 1-qubit unitary by a
short product of {HAD, T} using a spaceefficient version of the Solovay-Kitaev
theorem, recently proved by [vM,W].
Altogether:
Given A:
Run phase estimation with U=eiA on the
completely mixed state.
This uniformly sample an approximation of
an (eigenvector, eigenvalue) pair, in
logarithmic space.
Approximating the whole
spectrum
First attempt: Repeated
sampling
• Assume all eigenvalues are in [-1,1] .
• Divide [-1,1] to small consecutive intervals.
• For each interval, pick poly(n) independent
samples, and estimate the number of
eigenvalues in the interval by the fraction of
samples that fall into it.
-1
1
A problem: eigenvalues close to
a boundary
Eigenvalues that lie close to an interval
boundary might fall into both neighboring
intervals and lead to wrong results.
-1
1
The solution: Consistent
estimation
A probabilistic/quantum algorithm estimates a
value z, if w.h.p. it outputs a value close to z.
A probabilistic/quantum algorithm consistently
estimates a value z, if w.h.p. it outputs a fixed
value close to z.
Consistent sampling solves the problem above.
Consistent Sampling using
the shift & truncate method [SZ]
Original accuracy: 2-10
2-10
Consistent Sampling using
the shift & truncate method [SZ]
Original accuracy: 2-10
New accuracy: 2-20
2-10
Pick uniformly a value 0 < k < 210 and fix it.
Shift the eigenvalues by the fixed shift k* 2-20
Now, w.h.p., all eigenvalues are far away from a boundary.
Approximate the spectrum using
consistent sampling
• Divide [-1,1] to small consecutive intervals.
• For each interval, pick poly(n) independent
samples, and estimate the number of
eigenvalues in the interval by the fraction of
samples that fall into it.
-1
1
Approximating the
eigenvectors
Quantum state tomography
Quantum tomography is the process of
reconstructing the quantum state for a
source by measurements on the systems
coming from the source.
Quantum tomography is possible if we can
repeatedly and consistently generate the
same state.
Estimating an eigenvector
We saw we can consistently estimate an
eigenvalue i. Each time we get i we have
the n-dimensional eigenvector vi, represented
with log(n) qubits.
Using quantum state tomography we
efficiently output the n coordinates of vi.
Quantum tomography in small
space
For each k, l:
Where:
E (1) projects onto |k>,
E (2) projects onto |l>,
E (3) projects onto |k>+| l >, and,
E (4) projects onto |k>+i | l >,
Inverting a matrix
Inverting a matrix whose
eigenvalues are well separated.
• Approximate the eigenvalues,
D=Diag( 1,…, n)
• Approximate the eigenvectors v1,…,vn.
V=(v1,…,vn)
Then,
A VDV†
A-1 VD-1V†
Some reflections
BQL is surprisingly powerful
Either:
• BQL is indeed stronger than BPL, or
• BPL is also surprisingly powerful.
Reingold showed USTCON
L,
So far, this was not extended to RL=L.
An intriguing question : Can one approximately
invert stochastic matrices in BPL?