Quantum computing

The qubit serves as the basic unit of quantum information. It represents a two-state system, just like a classical bit, except that it can exist in a superposition of its two states. In one sense, a superposition is like a probability distribution over the two values. However, a quantum computation can be influenced by both values at once, inexplicable by either state individually. In this sense, a "superposed" qubit stores both values simultaneously.

A two-dimensional vector mathematically represents a qubit state. Physicists typically use Dirac notation for quantum mechanical linear algebra, writing |ψ⟩ 'ket psi' for a vector labeled ψ. Because a qubit is a two-state system, any qubit state takes the form α|0⟩ + β|1⟩, where |0⟩ and |1⟩ are the standard basis states,[lower-alpha 2] and α and β are the probability amplitudes. If either α or β is zero, the qubit is effectively a classical bit; when both are nonzero, the qubit is in superposition. Such a quantum state vector acts similarly to a (classical) probability vector, with one key difference: unlike probabilities, probability amplitudes are not necessarily positive numbers. Negative amplitudes allow for destructive wave interference.[lower-alpha 3]

When a qubit is measured in the standard basis, the result is a classical bit. The Born rule describes the norm-squared correspondence between amplitudes and probabilities—when measuring a qubit α|0⟩ + β|1⟩, the state collapses to |0⟩ with probability |α|², or to |1⟩ with probability |β|². Any valid qubit state has coefficients α and β such that |α|² + |β|² = 1. As an example, measuring the qubit 1/√2|0⟩ + 1/√2|1⟩ would produce either |0⟩ or |1⟩ with equal probability.

Each additional qubit doubles the dimension of the state space. As an example, the vector 1/√2|00⟩ + 1/√2|01⟩ represents a two-qubit state, a tensor product of the qubit |0⟩ with the qubit 1/√2|0⟩ + 1/√2|1⟩. This vector inhabits a four-dimensional vector space spanned by the basis vectors |00⟩, |01⟩, |10⟩, and |11⟩. The Bell state 1/√2|00⟩ + 1/√2|11⟩ is impossible to decompose into the tensor product of two individual qubits—the two qubits are entangled because their probability amplitudes are correlated. In general, the vector space for an n-qubit system is 2ⁿ-dimensional, and this makes it challenging for a classical computer to simulate a quantum one: representing a 100-qubit system requires storing 2¹⁰⁰ classical values.

The state of this one-qubit quantum memory can be manipulated by applying quantum logic gates, analogous to how classical memory can be manipulated with classical logic gates. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a matrix

$${\displaystyle X:={\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}$$

Mathematically, the application of such a logic gate to a quantum state vector is modelled with matrix multiplication. Thus

${\displaystyle X|0\rangle =|1\rangle }$ and ${\displaystyle X|1\rangle =|0\rangle }$.

The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way is simply to select a qubit and apply that gate to the target qubit while leaving the remainder of the memory unaffected. Another way is to apply the gate to its target only if another part of the memory is in a desired state. These two choices can be illustrated using another example. The possible states of a two-qubit quantum memory are

$${\displaystyle |00\rangle$$ :={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\quad |01\rangle :={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\quad |10\rangle :={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\quad |11\rangle :={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}.}

The CNOT gate can then be represented using the following matrix:

$${\displaystyle \operatorname {CNOT}$$ :={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}.}

As a mathematical consequence of this definition, ${\textstyle \operatorname {CNOT} |00\rangle =|00\rangle }$, ${\textstyle \operatorname {CNOT} |01\rangle =|01\rangle }$, ${\textstyle \operatorname {CNOT} |10\rangle =|11\rangle }$, and ${\textstyle \operatorname {CNOT} |11\rangle =|10\rangle }$. In other words, the CNOT applies a NOT gate (${\textstyle X}$ from before) to the second qubit if and only if the first qubit is in the state ${\textstyle |1\rangle }$. If the first qubit is ${\textstyle |0\rangle }$, nothing is done to either qubit.

In summary, a quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements.

Quantum parallelism refers to the ability of quantum computers to evaluate a function for multiple input values simultaneously. This can be achieved by preparing a quantum system in a superposition of input states, and applying a unitary transformation that encodes the function to be evaluated. The resulting state encodes the function's output values for all input values in the superposition, allowing for the computation of multiple outputs simultaneously. This property is key to the speedup of many quantum algorithms.[18]

There are a number of models of computation for quantum computing, distinguished by the basic elements in which the computation is decomposed.

A quantum circuit diagram implementing a Toffoli gate from more primitive gates

A quantum gate array decomposes computation into a sequence of few-qubit quantum gates. A quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements.

Any quantum computation (which is, in the above formalism, any unitary matrix of size ${\displaystyle 2^{n}\times 2^{n}}$ over ${\displaystyle n}$ qubits) can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set, since a computer that can run such circuits is a universal quantum computer. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem.

A measurement-based quantum computer decomposes computation into a sequence of Bell state measurements and single-qubit quantum gates applied to a highly entangled initial state (a cluster state), using a technique called quantum gate teleportation.

An adiabatic quantum computer, based on quantum annealing, decomposes computation into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution.[41]

A topological quantum computer decomposes computation into the braiding of anyons in a 2D lattice.[42]

A quantum Turing machine is the quantum analog of a Turing machine.[8] All of these models of computation—quantum circuits,[43] one-way quantum computation,[44] adiabatic quantum computation,[45] and topological quantum computation[46]—have been shown to be equivalent to the quantum Turing machine; given a perfect implementation of one such quantum computer, it can simulate all the others with no more than polynomial overhead. This equivalence need not hold for practical quantum computers, since the overhead of simulation may be too large to be practical.

A notable application of quantum computation is for attacks on cryptographic systems that are currently in use. Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers (e.g., products of two 300-digit primes).[53] By comparison, a quantum computer could solve this problem exponentially faster using Shor's algorithm to find its factors.[54] This ability would allow a quantum computer to break many of the cryptographic systems in use today, in the sense that there would be a polynomial time (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.

Identifying cryptographic systems that may be secure against quantum algorithms is an actively researched topic under the field of post-quantum cryptography.[55][56] Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory.[55][57] Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem.[58] It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2^n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2ⁿ in the classical case,[59] meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search (see Key size).

The most well-known example of a problem that allows for a polynomial quantum speedup is unstructured search, which involves finding a marked item out of a list of ${\displaystyle n}$ items in a database. This can be solved by Grover's algorithm using ${\displaystyle O({\sqrt {n}})}$ queries to the database, quadratically fewer than the ${\displaystyle \Omega (n)}$ queries required for classical algorithms. In this case, the advantage is not only provable but also optimal: it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups.

Problems that can be efficiently addressed with Grover's algorithm have the following properties:[60][61]

1. There is no searchable structure in the collection of possible answers,
2. The number of possible answers to check is the same as the number of inputs to the algorithm, and
3. There exists a boolean function that evaluates each input and determines whether it is the correct answer

For problems with all these properties, the running time of Grover's algorithm on a quantum computer scales as the square root of the number of inputs (or elements in the database), as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied[62] is Boolean satisfiability problem, where the database through which the algorithm iterates is that of all possible answers. An example and possible application of this is a password cracker that attempts to guess a password. Breaking symmetric ciphers with this algorithm is of interest to government agencies.[63]

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, quantum simulation may be an important application of quantum computing.[64] Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.[65]

About 2% of the annual global energy output is used for nitrogen fixation to produce ammonia for the Haber process in the agricultural fertilizer industry (even though naturally occurring organisms also produce ammonia). Quantum simulations might be used to understand this process and increase the energy efficiency of production.[66]

Quantum annealing relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which slowly evolves to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process. Adiabatic optimization may be helpful for solving computational biology problems.[67]

Since quantum computers can produce outputs that classical computers cannot produce efficiently, and since quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up machine learning tasks.[68][69]

For example, the quantum algorithm for linear systems of equations, or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts.[70][69] Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines and deep neural networks.[71][72][73]

Deep generative chemistry models emerge as powerful tools to expedite drug discovery. However, the immense size and complexity of the structural space of all possible drug-like molecules pose significant obstacles, which could be overcome in the future by quantum computers. Quantum computers are naturally good for solving complex quantum many-body problems[74] and thus may be instrumental in applications involving quantum chemistry. Therefore, one can expect that quantum-enhanced generative models[75] including quantum GANs[76] may eventually be developed into ultimate generative chemistry algorithms.

There are a number of technical challenges in building a large-scale quantum computer.[77] Physicist David DiVincenzo has listed these requirements for a practical quantum computer:[78]

• Physically scalable to increase the number of qubits
• Qubits that can be initialized to arbitrary values
• Quantum gates that are faster than decoherence time
• Universal gate set
• Qubits that can be read easily.

Sourcing parts for quantum computers is also very difficult. Superconducting quantum computers, like those constructed by Google and IBM, need helium-3, a nuclear research byproduct, and special superconducting cables made only by the Japanese company Coax Co.[79]

The control of multi-qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This has led to the development of quantum controllers that enable interfacing with the qubits. Scaling these systems to support a growing number of qubits is an additional challenge.[80]

One of the greatest challenges involved with constructing quantum computers is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T₂ (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.[81] Currently, some quantum computers require their qubits to be cooled to 20 millikelvin (usually using a dilution refrigerator[82]) in order to prevent significant decoherence.[83] A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.[84]

As a result, time-consuming tasks may render some quantum algorithms inoperable, as attempting to maintain the state of qubits for a long enough duration will eventually corrupt the superpositions.[85]

These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time.

As described in the threshold theorem, if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often-cited figure for the required error rate in each gate for fault-tolerant computation is 10⁻³, assuming the noise is depolarizing.

Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L², where L is the number of digits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 10⁴ bits without error correction.[86] With error correction, the figure would rise to about 10⁷ bits. Computation time is about L² or about 10⁷ steps and at 1 MHz, about 10 seconds. However, other careful estimates[87][88] lower the qubit count to 3 million for factorizing 2,048-bit integer in 5 months on a trapped-ion quantum computer.

Another approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads, and relying on braid theory to form stable logic gates.[89][90]

Quantum supremacy is a term coined by John Preskill referring to the engineering feat of demonstrating that a programmable quantum device can solve a problem beyond the capabilities of state-of-the-art classical computers.[91][92][93] The problem need not be useful, so some view the quantum supremacy test only as a potential future benchmark.[94]

In October 2019, Google AI Quantum, with the help of NASA, became the first to claim to have achieved quantum supremacy by performing calculations on the Sycamore quantum computer more than 3,000,000 times faster than they could be done on Summit, generally considered the world's fastest computer.[95][96][97] This claim has been subsequently challenged: IBM has stated that Summit can perform samples much faster than claimed,[98][99] and researchers have since developed better algorithms for the sampling problem used to claim quantum supremacy, giving substantial reductions to the gap between Sycamore and classical supercomputers[100][101][102] and even beating it.[103][104][105]

In December 2020, a group at USTC implemented a type of Boson sampling on 76 photons with a photonic quantum computer, Jiuzhang, to demonstrate quantum supremacy.[106][107][108] The authors claim that a classical contemporary supercomputer would require a computational time of 600 million years to generate the number of samples their quantum processor can generate in 20 seconds.[109]

On November 16, 2021, at the quantum computing summit, IBM presented a 127-qubit microprocessor named IBM Eagle.[110]

Some researchers have expressed skepticism that scalable quantum computers could ever be built, typically because of the issue of maintaining coherence at large scales, but also for other reasons.

Bill Unruh doubted the practicality of quantum computers in a paper published in 1994.[111] Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.[112] Skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved.[113][114][115] Physicist Mikhail Dyakonov has expressed skepticism of quantum computing as follows:

"So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be... about 10³⁰⁰... Could we ever learn to control the more than 10³⁰⁰ continuously variable parameters defining the quantum state of such a system? My answer is simple. No, never."[116][117]

For physically implementing a quantum computer, many different candidates are being pursued, among them (distinguished by the physical system used to realize the qubits):

• Superconducting quantum computing[118][119] (qubit implemented by the state of nonlinear resonant superconducting circuits containing Josephson junctions)
• Trapped ion quantum computer (qubit implemented by the internal state of trapped ions)
• Neutral atoms in optical lattices (qubit implemented by internal states of neutral atoms trapped in an optical lattice)[120][121]
• Quantum dot computer, spin-based (e.g. the Loss-DiVincenzo quantum computer[122]) (qubit given by the spin states of trapped electrons)
• Quantum dot computer, spatial-based (qubit given by electron position in double quantum dot)[123]
• Quantum computing using engineered quantum wells, which could in principle enable the construction of a quantum computer that operates at room temperature[124][125]
• Coupled quantum wire (qubit implemented by a pair of quantum wires coupled by a quantum point contact)[126][127][128]
• Nuclear magnetic resonance quantum computer (NMRQC) implemented with the nuclear magnetic resonance of molecules in solution, where qubits are provided by nuclear spins within the dissolved molecule and probed with radio waves
• Solid-state NMR Kane quantum computer (qubit realized by the nuclear spin state of phosphorus donors in silicon)
• Vibrational quantum computer (qubits realized by vibrational superpositions in cold molecules)[129]
• Electrons-on-helium quantum computer (qubit is the electron spin)
• Cavity quantum electrodynamics (CQED) (qubit provided by the internal state of trapped atoms coupled to high-finesse cavities)
• Molecular magnet[130] (qubit given by spin states)
• Fullerene-based ESR quantum computer (qubit based on the electronic spin of atoms or molecules encased in fullerenes)[131]
• Nonlinear optical quantum computer (qubits realized by processing states of different modes of light through both linear and nonlinear elements)[132][133]
• Linear optical quantum computer (LOQC) (qubits realized by processing states of different modes of light through linear elements e.g. mirrors, beam splitters and phase shifters).[134] Quantum microprocessor based on laser photonics at room temperature made possible.[135][136]
• Diamond-based quantum computer[137][138][139][140] (qubit realized by the electronic or nuclear spin of nitrogen-vacancy centers in diamond)
• Bose-Einstein condensate-based quantum computer[141][142]
• Transistor-based quantum computer (string quantum computers with entrainment of positive holes using an electrostatic trap)
• Rare-earth-metal-ion-doped inorganic crystal based quantum computer[143][144] (qubit realized by the internal electronic state of dopants in optical fibers)
• Metallic-like carbon nanospheres-based quantum computer[145]

The large number of candidates demonstrates that quantum computing, despite rapid progress, is still in its infancy.[146]

Any computational problem solvable by a classical computer is also solvable by a quantum computer.[2] Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using quantum mechanics, which underlies the operation of quantum computers.

Conversely, any problem solvable by a quantum computer is also solvable by a classical computer. It is possible to simulate both quantum and classical computers manually with just some paper and a pen, if given enough time. More formally, any quantum computer can be simulated by a Turing machine. In other words, quantum computers provide no additional power over classical computers in terms of computability. This means that quantum computers cannot solve undecidable problems like the halting problem, and the existence of quantum computers does not disprove the Church–Turing thesis.[147]

While quantum computers cannot solve any problems that classical computers cannot already solve, it is suspected that they can solve certain problems faster than classical computers. For instance, it is known that quantum computers can efficiently factor integers, while this is not believed to be the case for classical computers.

The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with an error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be solved by polynomial-time probabilistic Turing machines with bounded error.[148] It is known that ${\displaystyle {\mathsf {BPP\subseteq BQP}}}$ and is widely suspected that ${\displaystyle {\mathsf {BQP\subsetneq BPP}}}$, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.[149]

The suspected relationship of BQP to several classical complexity classes[50]

The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that ${\displaystyle {\mathsf {P\subseteq BQP\subseteq PSPACE}}}$; that is, all problems that can be efficiently solved by a deterministic classical computer can also be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer can also be solved by a deterministic classical computer with polynomial space resources. It is further suspected that BQP is a strict superset of P, meaning there are problems that are efficiently solvable by quantum computers that are not efficiently solvable by deterministic classical computers. For instance, integer factorization and the discrete logarithm problem are known to be in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known beyond the fact that some NP problems that are believed not to be in P are also in BQP (integer factorization and the discrete logarithm problem are both in NP, for example). It is suspected that ${\displaystyle {\mathsf {NP\nsubseteq BQP}}}$; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of NP-complete problems (if an NP-complete problem were in BQP, then it would follow from NP-hardness that all problems in NP are in BQP).[150]

The relationship of BQP to the basic classical complexity classes can be summarized as follows:

${\displaystyle {\mathsf {P\subseteq BPP\subseteq BQP\subseteq PP\subseteq PSPACE}}}$

It is also known that BQP is contained in the complexity class ${\displaystyle \color {Blue}{\mathsf {\#P}}}$ (or more precisely in the associated class of decision problems ${\displaystyle {\mathsf {P^{\#P}}}}$),[150] which is a subclass of PSPACE.

It has been speculated that further advances in physics could lead to even faster computers. For instance, it has been shown that a non-local hidden variable quantum computer based on Bohmian Mechanics could implement a search of an N-item database in at most ${\displaystyle O({\sqrt[{3}]{N}})}$ steps, a slight speedup over Grover's algorithm, which runs in ${\displaystyle O({\sqrt {N}})}$ steps. Note, however, that neither search method would allow quantum computers to solve NP-complete problems in polynomial time.[151] Theories of quantum gravity, such as M-theory and loop quantum gravity, may allow even faster computers to be built. However, defining computation in these theories is an open problem due to the problem of time; that is, within these physical theories there is currently no obvious way to describe what it means for an observer to submit input to a computer at one point in time and then receive output at a later point in time.[152][153]