What is a quantum computer? Quantum Architecture by Murat Birben 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. 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. 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 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 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. 1
Schrödinger's cat 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? Copenhagen interpretation 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 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 Multiple Qubits 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> = i 1 i 2...i n > = i 1 > XOR... XOR i n > Quantum Gates 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. Basic quantum gates and their matrix representations. 2
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? 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. EPR Paradox 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: 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. In addition, Preskill lists other elements necessary for fault tolerant computation in order to maintain a reasonable accuracy threshold. Two of these are 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. 3
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 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. SOLID-STATE TECHNOLOGIES (contd.) 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. Transporting Quantum Information 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 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. Swapping Channel (contd.) 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. 4
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. 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 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. 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. Architecture for a Quantum Wire 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. 5
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 10 15 to 10 9 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 Overview of Quantum Fault- Tolerant 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 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. 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 No single gate failure can lead to more than one error in each encoded qubit block. The impact of this requirement: 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. Recursive Error Correction 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. 6
ERROR-CORRECTION ALGORITHMS 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. [7,1,3] Code A parity measurement consists of the following steps. 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, A 0 >, 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. 7