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Quantum Computing Joseph C Quantum Computing Joseph C. Bardin, Daniel Sank, Ofer Naaman, and Evan Jeffrey ©ISTOCKPHOTO.COM/SOLARSEVEN uring the past decade, quantum com- underway at many companies, including IBM [2], Mi- puting has grown from a field known crosoft [3], Google [4], [5], Alibaba [6], and Intel [7], mostly for generating scientific papers to name a few. The European Union [8], Australia [9], to one that is poised to reshape comput- China [10], Japan [11], Canada [12], Russia [13], and the ing as we know it [1]. Major industrial United States [14] are each funding large national re- Dresearch efforts in quantum computing are currently search initiatives focused on the quantum information Joseph C. Bardin ([email protected]) is with the University of Massachusetts Amherst and Google, Goleta, California. Daniel Sank ([email protected]), Ofer Naaman ([email protected]), and Evan Jeffrey ([email protected]) are with Google, Goleta, California. Digital Object Identifier 10.1109/MMM.2020.2993475 Date of current version: 8 July 2020 24 1527-3342/20©2020IEEE August 2020 Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. sciences. And, recently, tens of start-up companies have Quantum computing has grown from emerged with goals ranging from the development of software for use on quantum computers [15] to the im- a field known mostly for generating plementation of full-fledged quantum computers (e.g., scientific papers to one that is Rigetti [16], ION-Q [17], Psi-Quantum [18], and so on). poised to reshape computing as However, despite this rapid growth, because quantum computing as a field brings together many different we know it. disciplines, there is currently a shortage of engineers who understand both the engineering aspects (e.g., mi- a0 ||}HH==aa0101+ |,H (1) crowave design) and the quantum aspects required to ;a1E build a quantum computer [19]. The aim of this article is to introduce microwave engineers to quantum com- where |0 H and |1H are the zero and one computa- puting and demonstrate how the microwave commu- tional-basis vectors of the qubit, respectively, and the nity’s expertise could contribute to that field. complex coefficients a0 and a1 are referred to as proba- To begin, we pose the question, What’s the big deal bility amplitudes. Physically, the probability amplitudes about quantum computing anyway? To put things in describe the likelihood that one would find the qubit perspective, let’s start by considering a regular digi- in the associated state were a measurement to be car- 2 2 tal computer—or, in the language of quantum engi- ried out: P ||0H = a0| and P |.1H =||a1 Note that in " , " , neers, a classical computer. The fundamental unit of (1) we have implicitly introduced Dirac notation, where information in a computer is the bit. In a conventional each of our computational-basis vectors is represented computer, one bit of memory is stored in the charge by a ket (i.e., |0 H and |)H .1 on a dynamic random-access memory (DRAM) cell. A quantum register comprises an ensemble of qubits, A memory register with N bits has 2N possible states and, just as with the digital register, an N-qubit regis- in the range {,00ff00 01,,01ff01ff,}11 . Thus, a ter has 2N possible states. However, since qubits can 300-b memory can store any of 2300 possible bit strings, exist in superposition states, the instantaneous state of a set so large that it is physically impossible to print the quantum register can be in any superposition of its N T it out since there is not enough matter in the universe 2 basis states; that is, |}aH ==00ff00aa01 f 11f 1 6 @ to make the ink, not to mention the paper (the infor- aa00ff00||00ff00HH++01 01 gf+ a11f 1|,111H where mation capacity of the observable universe, if every the probability amplitudes, ai, are constrained so that 2 2 2 degree of freedom of every particle was used to encode ||a00f 0 + ||aa00ff1 ++g ||111 = .1 Similar to the case a bit, is roughly 2b300 [20]). However, while a 300-b reg- of a single qubit, each probability amplitude describes ister can assume any of its 2300 possible states, we know the likelihood that the associated bit string would be that each bit is either zero or one, so at any given time measured the associated bit string if the state of the the register can store only one of the bit strings. quantum register were measured. In a classical computer, computations are carried out Computation on a quantum computer is typically by moving bits into a CPU, where logical operations are carried out as an in-memory operation. That is, rather implemented in physical circuitry. The essential func- than moving data from the memory to a CPU and back, tion of the CPU is to provide a prescribed output bit the quantum register is operated upon directly. An string for each possible input bit string so as to carry out operator that can be applied to a quantum register is a desired logical operation. Since this is digital circuitry, described by a unitary matrix t ,U which preserves the we know that for each input bit string the processor length of the unit state vector (UUtt@ = I). A universal will produce exactly one output bit string and that this quantum computer permits the application of any arbi- input-to-output mapping is defined in hardware, per- trary unitary operation to the quantum register; i.e., a haps with some subset of the input bits determining the universal quantum computer can perform any trans- logical operation applied to the remainder of the bits. formation of the form In a quantum computer, we still have physical objects representing bits, but the difference is that the ||}}lHH= t ,U (2) memory register exists as a quantum state. The basic building block in a quantum register is the quantum where Ut is an arbitrary unitary matrix. bit, or qubit. Just like a regular bit, a qubit has a zero Unitary operators should be nothing new to a micro- (ground) state and a one (excited) state. However, what wave engineer; the scattering matrix of a lossless passive distinguishes a qubit from a classical bit is that it can be network is unitary (SS@ = I). In fact, from a mathemati- in a superposition of its zero and one states. Mathemat- cal perspective, the way a quantum computation acts ically, the state of a qubit is described by a 2D complex upon the 2N states of a quantum register is identical to state vector of unit amplitude, the operation that occurs when N incident waves scatter August 2020 25 Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on October 01,2020 at 19:47:20 UTC from IEEE Xplore. Restrictions apply. from a lossless N-port network (see Figure 1). Therefore, |.}HH=+05()||00 01HH++|| 10 11H , which can be real- we can see that familiar microwave concepts, such as ized by two isolated qubits, may be “factored” into superposition and interference, are also useful in pro- two single-qubit states |.}HH=+05()||01HH# (| 01+|)H . viding intuition with respect to quantum computing; On the other hand, the Bell state |(}HH=+||H)0011/ 2 complex microwave signal amplitudes are analogous cannot be factored (try it!), so it cannot be thought of as to complex quantum probability amplitudes, and, representing the information of two independent bits. whereas the signals in Figure 1(a) carry power, those in States that cannot be thought of as combinations of Figure 1(b) carry probabilities. single-qubit states are called entangled, and the genera- There is an important distinction between the tion of such states is essential in accessing the power of physical N-port model and the conceptual model of quantum computing. quantum computation: while the microwave N-port While universal quantum computing permits op­­ has one physical port associated with each input/out- eration on all 2N computational states of a quantum put wave, the equivalent diagram describing quantum register, we cannot perform a measurement of a quan- computation has a port associated with each of its 2N tum superposition state. Instead, when we mea- computational states. Thus, the computational space of sure the state of the quantum register at the end of a universal quantum computer grows exponentially a computation, the computer must make a decision. with the number of qubits. This massive computational As dictated by the postulates of quantum mechan- capacity explains why quantum computing is so prom- ics, such a measurement returns a single classical bit ising: even at roughly 50 qubits, the computational of information per measured qubit, meaning that, if space is too large to simulate using the best supercom- one measures the state of an N-qubit quantum reg- puters, while an ideal quantum computer can easily ister, one will find the register in one of its 2N basis operate across the entire expanse [5]. states, with the likelihood of each outcome equal to Although superposition and interference make the modulus of the associated probability amplitude 2 2 qubits a little more complex than classical bits, those (i.e., P ||00f 0H = a00f 0|, P ||00f 1H = a00f 1|, and " , " , features alone do not provide the exponential com- so on). Additionally, the act of measuring the state putational space described previously. For instance, of a quantum register will cause any superposition a set of N isolated qubits is only about as complex as state to collapse, and the best one can do is that the a set of N classical bits because, when the qubits oper- postmeasurement state of the register agrees with the ate in isolation, only a small subset of possible super- measurement result.
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