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What is a quantum computer?
Quantum Architecture
by
Murat Birben
What is a Quantum Computer
(contd.)
• At any time the computer can be in a
superposition of all possible machine
states. This gives quantum computing the
potential power of proceeding through all
computational paths simultaneously.
• Between measurements, the machine
state evolves unitarily. This restricts the
power of quantum computing. Only
reversible computations are possible.
• A quantum computer is a device designed to
take advantage of distincly quantum phenomena
in carrying out a computational task.
• A quantum system can be put in a superposition
of multiple states and can evolve along multiple
trajectories simultaneously, only to choose a
definite state at the time of measurement.
• A quantum computer tries to exploit this
parallelism that classical computers do not
utilize.
Quantum Mechanics
• Quantum mechanics is a probabilistic theory, and it is
this randomness that places limitations on the accuracy
of characterizing a system.
• Let us consider a particle, say an electron, moving
through space. We describe the electron's motion in
terms of its position and momentum.
• Classically we can measure both quantities to infinite
precision.
• However in Quantum Mechanics we can never know
both quantities absolutely precisely.
• This is because by taking a measurement we
inadvertantly disturb the system as well.
Quantum Superposition
Quantum Entanglement
• Quantum superposition is the fundamental law
of quantum kinematics. It defines the allowed
state space of a quantum mechanical system.
• The principle of superposition states that if the
world can be in any configuration, any possible
arrangement of particles or fields, and if the
world could also be in another configuration,
then the world can also be in a state which is a
mixture of the two, where the amount of each
configuration that is in the mixture is specified by
a complex number.
• Quantum entanglement is a quantum
mechanical phenomenon in which the
quantum states of two or more objects are
linked together so that one object cannot
be adequately described without full
mention of its counterpart — even though
the individual objects may be spatially
separated.
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Schrödinger's cat
Copenhagen interpretation
• Schrödinger's famous thought experiment poses
the question: when does a quantum system stop
existing as a mixture of states and become one
or the other?
• In the Copenhagen interpretation of
quantum mechanics, a system stops being
a superposition of states and becomes
either one or the other when an
observation takes place.
Qubit
Multiple Qubits
• The basic information bearing element of
quantum computation is qubit(quantum
bit).
• Qubit is a two-dimensional quantum
system
– |Ψ> = α|0> + β|1>
• Where
– |α|2 + |β|2 = 1
Quantum Gates
• For two qubits, the state space is 4
dimensional
– |0> = |00> = |0> XOR |0>
– |1> = |01> = |0> XOR |1>
– |2> = |10> = |1> XOR |0>
– |3> = |11> = |1> XOR |1>
• For n qubits, state space is 2ⁿ dimensional
with basis vectors
– |i> = |i1i2...in> = |i1> XOR ... XOR |in>
Basic quantum gates and their
matrix representations.
• A collection of n qubits is called a quantum
register of size n.
• Quantum gates are the functional building
blocks of a quantum computer.
• A single gate in a quantum circuit with one or
more input qubits in the initial state |Ψ>
transforms the state to a different state | Ψ`> by
changing all probability amplitudes that describe
• the state vector [c 0, c 1, . . . , c n−1]T.
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Basic Quantum Gates (contd.)
• The classical NOT has the quantum analogue X which
inverts the probabilities of measuring 0 and 1.
• The quantum analogue of XOR is the two-qubit CNOT
gate: the target qubit is inverted for those states where
the source qubit is 1.
• Most quantum gates, however, have no classical
analogue.
• The combination of T, H, and CNOT provide a universal
set: just as any Boolean circuit can be composed from
AND, OR, and NOT gates, any polynomially describable
multiqubit quantum transform can be efficiently
approximated by composing just these three quantum
gates into a circuit.
• A quantum SWAP can be implemented as three CNOTs.
Quantum Teleportation
• Quantum teleportation is the recreation of
a quantum state at a distance, using only
classical communication.
Why bother with teleportation?
EPR Paradox
• First, we can precommunicate EPR pairs
with extensive pipelining without stalling
computations.
• Second, it is easier to transport EPR pairs
than real data.
• Third, transmitting the two classical bits
resulting from the measurements is more
reliable than transmitting quantum data.
• In quantum mechanics, the EPR paradox is a
thought experiment which challenged long-held
ideas about the relation between the observed
values of physical quantities and the values that
can be accounted for by a physical theory
• The EPR paradox draws on a phenomenon
predicted by quantum mechanics, known as
quantum entanglement, to show that
measurements performed on spatially separated
parts of a quantum system can apparently have
an instantaneous influence on one another.
• Basic elements sufficient to build a scalable
quantum computer have been described by
DiVincenzo and Preskill. The five DiVincenzo
criteria for building a quantum computer are:
• In addition, Preskill lists other elements
necessary for fault tolerant computation in order
to maintain a reasonable accuracy threshold.
Two of these are
– a scalable physical system with well characterized
qubits
– the ability to initialize the state of the qubits to a
simple fiducial state
– long relevant decoherence times
– a universal set of quantum gates
– a qubit specific measurement capability.
– maximal parallelism and
– gates that can act on any pair of qubits.
• DiVincenzo mentions two additional criteria
essential for quantum communications namely:
– the ability to interconvert stationary and flying qubits
– the ability to faithfully transmit flying qubits between
specified locations.
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Toward a Scalable, Silicon-Based
Quantum
Computing Architecture
• Propose quantum teleportation as a
means to communicate data over longer
distances on a chip.
• In particular, transporting quantum data is
a critical requirement for upcoming siliconbased quantum computing technologies.
SOLID-STATE TECHNOLOGIES
(contd.)
SOLID-STATE TECHNOLOGIES
• The key feature of these solid-state platforms are as
follows.
– Quantum bits are laid out in silicon in a two-dimensional (2-D)
fashion, similar to traditional CMOS VLSI.
– Quantum interactions are near-neighbor between bits.
– Quantum bits cannot move physically, but quantum data can be
swapped between neighbors.
– The control structures necessary to manipulate the bits prevent a
dense 2-D grid of bits. Instead, we have linear structures of bits
which can cross, but there is a minimum distance between such
intersections that is on the order of 20 bits for our primary
technology model. This restriction is similar to a “design rule” in
traditional CMOS VLSI.
Transporting Quantum Information
• All quantum computing proposals, uses classical
signals to control the timing and sequence of
operations. All known quantum algorithms,
including basic error correction for quantum
data, require the determinism and reliability of
classical control. Without efficient classical
control, fundamental results demonstrating the
feasibility of quantum computation do not apply
• Transferring quantum states between atomicand photon-based technologies is currently
extremely difficult.
• One of the most important distinctions
between quantum and classical wires
arises from the no-cloning theorem is that
quantum information cannot be copied but
must rather be transported from source to
destination
Short Wires: Swapping Channel
Swapping Channel (contd.)
• In solid-state technologies, a line of qubits is one
plausible approach to transporting quantum data. Figure
below provides a schematic of a swapping channel in
which information is progressively swapped between
pairs of qubits in the quantum datapath—somewhat like
a bubble sort.
• A quantum swapping channel presents
significant technical challenges.
– The placement of the phosphorus atoms
themselves.
– The scale of classical control.
– The temperature of the device.
• Producing a line of quantum bits that
overcomes all of the above challenges is
possible.
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Long Wires: Teleportation Channel
• This architecture makes use of the
quantum primitive of teleportation.
• Three important architectural building
blocks:
– The entropy exchange unit
– The EPR generator
– The purification unit.
EPR Generator
• Constructing an EPR pair of qubits is straightforward.
• We start with two state qubits from our entropy exchange
unit. End up with a two-qubit entangled state of 1/sqrt(2)
(|00> + |11>): an ERP pair.
• The input is directly piped from the entropy exchange
unit and the output is the entangled EPR pair.
Architecture for a Quantum Wire
Entropy Exchange Unit
• The physics of quantum computation requires
that operations are reversible and conserve
energy.
• The entropy of the system increases through
qubits coupling with the external environment.
• The process can be viewed as one of
thermodynamic cooling.
• “Cool” qubits are distributed throughout the
processor, analogous to a ground plane in a
conventional CMOS chip. The “cool” qubits are
in a nearly zero state. They are created by
measuring the qubit, and inverting if |1> .
EPR Purification Unit
• This unit takes as input n EPR pairs, which have been
partially corrupted by errors, and outputs nE
asymptotically perfect EPR pairs.
• E is the entropy of entanglement, a measure of the
number of quantum errors which the pairs suffered.
• The quantum inputs to this block are the input EPR
states and a supply of |0> qubits.
• The outputs are pure EPR states.
• Note that the block is carefully designed to correct only
up to a certain number of errors; if more errors than this
threshold occur, then the unit fails with increasing
probability.
Analysis of the Teleportation
Channel
• The bandwidth of a teleportation channel is proportional
to the speed with which reliable EPR pairs are
communicated.
• Efficiency of the purification process must be taken into
account.
• The overall bandwidth of this long quantum wire is less
than the simple wiring scheme, but the decoherence
introduced does not change with the wire length which
makes an important difference.
• Unlike the short wire, this bandwidth is not constrained
by a maximum distance related to the Threshold
Theorem since teleportation is unaffected by distance.
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Fault Tolerant Architecture
• Key system requirement for quantum computing.
• The ability to tolerate and dynamically handle internal
faults while preserving the integrity of the computation.
• Unlike present classical computing systems, where the
gate failure probability is extremely low current and
projected quantum gates have to probabilities of 1015 to
109 times more failure per operation.
• Key to this study of quantum wires was a tradeoff
between the geometric design of the system and the
noise generated during operation
Quantum Error Correction
• The only errors which can occur to a classical bit are bitflips and erasures, which can be modeled as conditional
and random NOT gates.
• Quantum bits suffer more kinds of error, because of the
greater degree of freedom in their state representation;
surprisingly, however, there are general strategies for
reducing the universe of possible quantum errors to only
two kinds: bit-flips (random X gates) and phase-flips
(random Z gates).
• Quantum states collapse upon measurement, so
strategies must be employed for determining errors
without actually measuring encoded data.
Overview of Quantum FaultTolerant Strategy
• In order for a system to operate reliably despite
a partial corruption of the data it processes, it
must introduce a certain amount of redundancy
in the form of an error-correction code.
• It is possible to develop a fault-tolerant strategy
for quantum systems based on the recursive
encoding of states by concatenation of quantum
error-correction codes
Quantum Error Correction (contd.)
• Because of the no-cloning theorem,
quantum information cannot be simply
duplicated. Instead, redundancy is
achieved through entangled states with
known properties.
Computing on Encoded Data
Recursive Error Correction
• No single gate failure can lead to more than one
error in each encoded qubit block. The impact of
this requirement:
• A very simple construction allows us to
tolerate additional errors.
• If a logical qubit is encoded in a block of n
qubits, it is possible to encode each of
those qubits with an m-qubit code to
produce an mn encoding.
• Such recursion, or concatenation,of codes
can reduce the overall probability of error
even further.
– no single operation may cause multiple failures
– measurement errors must not be allowed to
propagate excessively.
• To achieve first one, no two encoding qubits are
allowed to both interact directly with a third qubit.
• To achieve second one, measurements are
performed in a multiple fashion.
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ERROR-CORRECTION
ALGORITHMS
[7,1,3] Code
• A parity measurement consists of the following steps.
• the notation [n,k,d] , where n is the number
of physical qubits, k is the number of
logical qubits encoded, and d is the
quantum Hamming distance of the code.
• A code with distance d is able to correct
(d-1)/2 errors.
– Prepare a cat state from four ancillae, using a Hadamard gate and three
CNOT gates.
– Verify the cat state by taking the parity of each pair of qubits. If any pair
has odd parity, return to step 1. This requires six additional ancillae, one
for each pair.
– Perform a CNOT between each of the qubits in the cat state and the
data qubits whose parity is to be measured
– Uncreate the cat state by applying the same operators used to create it
in reverse order. After applying the Hadamard gate to the final qubit,
|A0> , that qubit contains the parity.
– Measure |A0>:
• A With |A0> = α|0> +β|1>, create the three-qubit state α|000> +β|111> by
using |A0> as the control for two CNOT gates, and two fresh |0> ancillae as
the targets
• B Measure each of the three qubits.
– Use the majority measured value as the parity of the cat state.
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