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From page 24... ...
The mechanisms for manipulating classical and quantum bits are compared and contrasted to illustrate the unique challenges and benefits of quantum computing. The chapter concludes by describing the types of quantum computers currently being pursued by researchers, providing a first look at the progress that will be assessed in the chapters to follow.

From page 25... ...
For example, quantum objects can exist in multiple states all at once, with each of the states adding together and interfering like waves to define the overall quantum state. In general, the state of any quantum system is described in terms of "wave functions." In many cases, the state of a system can be expressed mathematically as a sum of the possible contributing states,2 each scaled by a complex number3 coefficient that reflects the relative weight of the state.

From page 26... ...
, formally called "measurement," occurs when the object interacts with some larger physical system that extracts information from it. Measurement fundamentally disrupts a quantum state: it "collapses" the aspect of wave function that was measured into a single observable state, resulting in a loss of information.

From page 27... ...
Research trends for quantum computing and algorithms, quantum communications, and quantum sensing and metrology are illustrated in Figure 2.1.6 The field of quantum information science generally explores how information can be encoded in a quantum system, including the associated statistics, limitations, and unique affordances of quantum mechanics. 6 See Appendix E for a discussion of research efforts by nation of origin.

From page 28... ...
While the evolution of the system follows a wave equation, any measurement of the system will return a value consistent with it being a particle. β’ Superposition  A quantum system can exist in two or more states at once, referred to as a "superposition" of states or a "superposition state." The wave function for such a superposition state can be described as a linear combination of the contributing states, with complex coefficients.

From page 29... ...
Quantum communication protocols are likely to be necessary for quantum computing  whether to transport information from one part of quantum computer hardware to another, or to enable communication between quantum computers. A subfield of quantum communication is quantum cryptography, in which quantum properties are used to design communication systems that may not be eavesdropped upon by an observer.7 7 The most prominent example is quantum key distribution (QKD)

From page 30... ...
Quantum sensors are commonly based upon qubits and are implemented using many of the same physical systems8 used in experimental quantum computers. Quantum computing, the primary focus of this report, leverages the quantum mechanical properties of interference, superposition, and entanglement to perform computations that are roughly analogous to (although they operate quite differently from)

From page 31... ...
For example, abstractions are used to reduce design complexity, which is essential when creating complex systems. Yet these abstractions often introduce systematic noise, since by hiding implementation details, the designers do not know the precise details of the implementation they are using.

From page 32... ...
for the digital inverter is called noise immunity. SOURCE: 2.2 Data generated using HSPICE, using 45 nm transistors models from the predictive technology modeling effort at Arizona State University; see Nanoscale Integra tion and Modeling (NIMO)

From page 33... ...
This translation allows today's hardware designers to describe their designs at a relatively high abstraction level and to use an automated design tool to map them to the required logic gates, a process called "logic synthesis." Since the number of basic building blocks is limited, all IC manufacturers provide a set of predesigned and tested logic gates, their "standard cell library," that may be incorporated into a chip's design and built in silicon using their manufacturing technology. Using both digital logic and standard libraries for these logic gates also makes designs robust  that is, they have negligible error rates.

From page 34... ...
These internal test points also enable tools to automatically generate tests that can confirm that the manufactured chip performs the exact same Boolean function as specified in the design, greatly simplifying manufacturing tests. As the next sections will show, while quantum computers have bitlike structures (called "qubits")

From page 35... ...
becomes one in the state that is read and zero in the other; all information about the amplitudes is destroyed upon measurement.10 Measurement outcomes for a single qubit are listed in Table 2.2 and explained in more detail in Box 2.2. 10 However, if one were to initialize a qubit in a specific state an arbitrary number of times, and measure it each time, one would be able to create a histogram of the number of times that a measurement yields each output, which would enable one to statistically approximate the relative probabilities associated with each state, and so infer the absolute value of the coefficient (equivalent to the square root of the calculated probability)

From page 36... ...
of System Premeasurement State ent Measurem Outcome Outcome Probability of State of System Postmeasurement ππ = = 01β© ππ 00β© Measurement of Outcome 100% ππ = = 01β© ππ 00β© Outcome ent 00 Outcome 100% State of System BOX 2.2 ππππ = 01β© = 00β© ππππ = = 00β© ππ= 01β© 00β© 00 01 100% a Single Qubit ππ = 01β© 1 1 qubit is in the state  Ο β© =  0β©,01 result of50% ππππ = 00β© = 01β© measurement will be 0 withππ = 11β© 00 01 00 100% 100% ππ = When a 00β© + 11β© 50% 1 1 a probability 2 100 percent, which is not11 ππ = 1 00β© + 1 11β© 2 ππππ = 00β© unlike what happens with a classical bit.= 11β© 00 100% the 11 50% ππ = 2 00β© + 2 11β© qubit in state11 β© =  1β© will 50% an outcome of 1 with = 11β© ππ of 00 50% 50% 2 2 Similarly, measurement of a Ο yield a probability of 100 percent. For a qubit in a superposition state, the outcome is less simple  the outcome of measurement, even of a known state, cannot be predicted with certainty.

From page 37... ...
This exponential scaling of the quantum state is what allows 32 qubits to represent all 232 possible outputs of a 32bit function and illustrates the richness of a quantum computer, and the difficulties in modeling these machines classically as they increase in size. This view also points out that, while qubits have "bit" in their name, they are neither digital nor purely binary.

From page 38... ...
2.4 COMPUTING WITH QUBITS The analog nature of qubit states and quantum gates dramatically changes the necessary design approaches and circuit architectures for quantum computers. (See Figure 2.3.)

From page 39... ...
, which provides user access to the quantum computer, and the needed software support services.

From page 40... ...
In addition to this difference in architecture, since quantum computers operate on different types of values than classical computers, they cannot use the same logical gate abstractions that were developed to manipulate classical bits. New abstractions for computations using qubits are required, providing a way to implement specified changes in quantum states.

From page 41... ...
2.4.1 Quantum Simulation, Quantum Annealing, and Adiabatic Quantum Computation Analog quantum computing involves a system of qubits in an initial quantum state, and changes to the Hamiltonian such that the problem is encoded in the final Hamiltonian and the final state corresponds to the answer. If the system remains in the ground state of the changing Hamiltonian, this approach is referred to as adiabatic quantum computing (AQC)

From page 42... ...
is performed by precisely changing the Hamiltonian of one or more qubits for the specific amount of time required to achieve the desired transformation. This is done by changing the physical environment, for example, via a laser pulse or application of some other electromagnetic field, depending on the way in which the qubits are built.14 Since these primitive operations are analogous to logic gates in classical computing, systems built using this approach are called "digital quantum computers." The rules of quantum mechanics constrain the set of possible quantum gate operations in a few interesting ways.

From page 43... ...
of Qubit Outcome of simpler primitive State of Qubit Outcome 1β© ππ = 1β© ππ = in particular, it is thus very important 100% methods which can = 1β© ππ cre 1 more easily be implemented in hardware. It is worth noting that known algorithms for some 0 0 ππ = 0β© applications, for example in computational chemistry, rely heavily upon general angle rota 1 1 1 1 1 tions; for these cases to have ππ = 0β© + 1β© ππ = 0β© + 1β© ππ = 1β© ate, or synthesize, these operations using a small number of primitive gate operations.

From page 44... ...
Ο Ο FIGURE 2.3.1β A picture of the Bloch sphere, which represents the set of all possible states for a single qubit. The qubit angles ΞΈ and Ο are shown in the figure.

From page 45... ...
QUANTUM COMPUTING: A NEW PARADIGM 45 FIGURE 2.4β Commonly used 1, 2, and 3qubit quantum gates, along with their FIGURE 2.4 corresponding unitary matrices, circuit symbols, and a description of their effects. The T, Hadamard, and CNOT gates are known to form a universal quantum gate set.

From page 46... ...
inside the quantum processor, like complex gates, can be hard to achieve,17 it can be decomposed into a number of simpler primitive gate operations directly supported by the hardware. This indirect coupling can be performed through a chain 16 If qubit A is entangled with qubit B, and at some later time qubit B becomes entangled with qubit C, it is likely that qubit A is now also entangled with qubit C

From page 47... ...
0 50% 1 50% ππ = 0β© 1 1 ππ = 0β© + 1β© ππ = 1β© 2 2 0 25% 1 75% ππ = 0β© 1 3 ππ = 0β© + 1β© ππ = 1β© 2 2 0 25% 1 75% ππ = 0β© 1 3ππ 3+4/6 TABLE 2.3β Measurement Outcomes and Probabilities for Some Possible States of a TwoQubit System, Given ππ = 1β© ππ = 0β© + 1β© 2 2 TABLE 2.3 Measurement Outcomes and Probabilities for Some Possible States of a TwoQubit System, Given Its Its Initial State Initial State Premeasurement State Premeasurement State Measurem Probability of Measurement Postmeasurement Probability of Postmeasurement State of (Wave Function) of System (Wave Function)

From page 48... ...
This condition is called "entanglement," and is inherently quantum mechanical. In mathematical terms, entanglement arises when there is no way to write the multiqubit wave function as the product of onequbit wave functions.

From page 49... ...
Since making and storing copies of intermediate states or partial results in memory is an essential part of classical computing and the way we think about programming, quantum computers require a different approach to algorithm design. Also, computing tasks often require the ability to access stored data, and many quantum algorithms require a means to access stored classical bits in a way that reveals which bits are being queried and loaded into quantum memory.

From page 50... ...
In addition to the fundamental qubit coherence errors, given the analog control signals used to perform qubit gate operations, each gate operation is not perfect, and the function for each possible input value. Yet measuring this quantum system directly will not yield this information.

From page 51... ...
When error rates are used as the gate fidelity metric, this rate accounts for the decoherence that occurs during the gate time, and any other errors caused by the gate operation. Given that the user of a quantum computer is interested in estimating the fidelity of the results, extracting effective gate error rates using the process of randomized benchmarking (RBM)

From page 52... ...
β’ Fully errorcorrected gatebased quantum computers. Such a sys tem would operate through gatebased operations on qubits, implementing quantum error correction to correct any system noise (including errors introduced by imperfect control signals or device fabrication, or unintended coupling of qubits to each other or to the environment)

From page 53... ...
In order to build a functional quantum computer, one must create a physical system that encodes qubits and control and manipulate these qubits precisely in order to carry out computations. Today, experimentalists are building and operating these systems in carefully controlled environments in laboratories.

From page 54... ...
The quantum logic gates on the qubits are carried out using the laser beams from the gate laser source, which is modulated by the control signals (RF signals delivered through the blue cables) and routed to the ions with the optical setup in the system.

From page 55... ...
BlumeKohout, 2013, Error suppres sion and error correction in adiabatic quantum computation: Techniques and chal lenges, Physical Review X 3:041013; A Mizel, 2014, "FaultTolerant, Universal Adiabatic Quantum Computation," https:// Β arxiv.org/abs/1403.7694; S.P.

From page 56... ...
Love, and C.J.S. Truncik, 2008, Thermally assisted adiabatic quantum computation, Physical Review Letters 100(6)

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