We often hear that quantum computers efficiently solve problems that are very difficult to solve with a classical computer. But even if the hardware is available to build a quantum computer, exploiting its quantum features requires us to write smart algorithms.

An algorithm is a sequence of logically connected mathematical steps that solve a problem. For example, an algorithm to add three numbers can have two steps: add the first two numbers in the first step and the result to the third number in the second step.

Quantum v. classical algorithms

A more involved example of an algorithm is the search for the largest number in a finite list of numbers.

An algorithm can start by assuming that the first number on the list is the largest.  Next, it can compare this number with the second number on the list. If the second number is larger than or equal to the first number, the second number is now deemed to be the largest.  Otherwise, the first number remains the largest at this stage. The algorithm then moves to the third number on the list – and so on until it has finished comparing all the numbers on the list. The number that is the largest as of the final step will be the answer.

A quantum algorithm is also a series of steps, but its implementation requires quantum gates. Some problems may need fewer steps on the part of a quantum algorithm than the number of steps required by a classical algorithm. That is, the quantum algorithm can speed up the computation.

One factor that controls this speed-up is the possibility of superposition of the states of quantum bits, or qubits, that encode information. Whereas a classical computer uses semiconductor-based gadgets as bits to encode information, quantum computers use qubits. In both cases, the bit or the qubit can have two distinct states, 0 or 1; but qubits have the additional ability to be partly 0 and partly 1 at the same time.

Shor’s algorithm

One of the earliest quantum algorithms is the factorisation algorithm developed by Peter Shor. It requires fewer steps to factorise a number than one that operates with classical principles.

Shor’s algorithm identifies the factors of a given integer. For example, 2 is a factor of 20 (since 2 divides 20 without a remainder). Similarly, 4, 5, and 10 are also factors of 20. However, identifying all the factors requires a greater and greater number of steps if the number becomes larger.

The efficiency of an algorithm is related to the number of steps required as the size of the input increases. An algorithm is more efficient if it requires fewer steps (and thus less time). From this perspective, Shor’s algorithm is far more efficient than any known classical algorithm for factorisation.

Technically, in Shor’s algorithm, the number of steps increases as a polynomial in the size (more precisely, the logarithm of the size) of the input whereas it is a superpolynomial for the best classical algorithm known today.

To understand the difference, compare multiplying 10 with itself thrice (i.e. 10^3) and multiplying 3 ten times (i.e. 3^10). The former is a polynomial in 10 whereas the latter is a  superpolynomial in 10. A polynomial increase is always lower than a superpolynomial increase for a sufficiently large input size. Thus, classical factorisation algorithms are far less efficient compared to Shor’s algorithm, which is a quantum algorithm.

Modern cryptography – which is used to secure user accounts on the internet, for example – depends on the fact that there are no efficient classical algorithms that can factorise large integers. This is the source of the claim that the availability of quantum computers (with an adequate number of qubits) will challenge the safety of classical cryptography.

Grover’s and Deutsch-Jozsa algorithms

Another popular quantum algorithm is the quantum search algorithm developed by Lov Grover. It looks for a numerical pattern in a large list of numbers. A deterministic classical algorithm requires almost half the number of steps as there are patterns in the list. That is, to identify a pattern from a list of one-million patterns, the classical approach may need half a million steps. The quantum algorithm will require only a thousand steps, however. In fact, for every 100x increase in the list’s size, Grover’s algorithm will need only 10x more steps. This is the kind of speed-up this quantum algorithm achieves.

Yet another scheme that showcases the exponential speed-up is the Deutsch-Jozsa algorithm. Imagine a set containing two-digit numbers whose digits are either 0 or 1; let’s call this Set A: 00, 01, 10, and 11. For each number from Set A, associate a number from another set, Set B, containing 0 and 1 as the only members.

Next, consider two categories of relation between the two sets. A relation is constant if all the members of the first set are associated with only 0 or only 1. A relation is balanced if two of the numbers from the first set are associated with 0 and the other two with 1.

Say the output is 0. A classical computer will require three steps at most to determine if the mapping is constant or balanced. (Can you figure out what they are?)

But a quantum computer can figure it out with only one computation. This is thanks to superposition – the ability of the value of a qubit to be partly 0 and partly 1 at the same time.

As this author wrote previously, “If a qubit is in a superposition, then measuring the qubit will cause it to collapse to one of the two states [either 0 or 1]. However, we can only predict the probability that it will collapse to one state.”

When the inputs are in superposition, the output will be as well, and in a way that corresponds to the states in the input superposition. The output will also have a sign – positive or negative – depending on whether the association is balanced or constant.

So the Deutsch-Jozsa algorithm can determine the mapping with one computation independent of the size of the input. We just need to make sure there are enough qubits available to represent the number of digits in the input. (Of course, this requirement would apply to bits as well).

Wait for reliable devices

Scientists already know of more quantum algorithms that can solve problems in optimisation, drug design, and pattern search, among other fields more efficiently.

When reliable, large-scale devices become available, quantum computing will help address many otherwise intractable problems as well. Research in quantum algorithms is highly interdisciplinary, involving computer science, mathematics, and physics. The field is also still evolving, and there are plenty of opportunities to make significant contributions.

S. Srinivasan is a professor of physics at Krea University.

Quantum computing is becoming more popular – both as a field of study and in the public imagination. The technology promises more speed and more efficient problem-solving abilities, challenging the boundaries set by classical, conventional computing.

The hype has led to inflated expectations. But whether or not it can meet them, the raison d’être of a quantum computer is taken to be synonymous with the ability to solve some problems much faster than a classical computer can. This achievement, called quantum supremacy, will establish quantum computers as superior machines.

Scientists have been exploring both experimental and theoretical ways to prove quantum supremacy.

Ramis Movassagh, a researcher at Google Quantum AI, recently had a study published in the journal Nature Physics. Here, he has reportedly demonstrated in theory that simulating random quantum circuits and determining their output will be extremely difficult for classical computers. In other words, if a quantum computer solves this problem, it can achieve quantum supremacy.

But why do such problems exist?

Facing the quantum challenge

Quantum computers use quantum bits, or qubits, whereas classical computers use binary bits (0 and 1). Qubits are fundamentally different from classical bits as they can have the value 0 or 1, as a classical bit can, or a value that’s a combination of 0 and 1, called a superposition.

Superposition states allow qubits to carry more information. This capacity for parallelism gives quantum computers their archetypal advantage over classical computers, allowing them to perform a disproportionately greater number of operations.

Qubits also exhibit entanglement, meaning that two qubits can be intrinsically linked regardless of their physical separation. This property allows quantum computers to tackle complex problems that may be out of reach of classical devices.

All this said, the real breakthrough in quantum computing is scalability. In classical computers, the processing power grows linearly with the number of bits. Add 50 bits and the processing power will increase by 50 units. So the more operations you want to perform, the more bits you add.

Quantum computers defy this linearity, however. When you add more qubits to a quantum computer, its computational power for certain tasks grows exponentially as 2ⁿ, where n is the number of qubits. For example, whereas a one-qubit quantum computer can perform 2¹ = 2 computations, a two-qubit quantum computer can perform 2² = 4 computations, and so forth.

#P-hard problems

Quantum circuits are at the heart of quantum computing. These circuits consist of qubits and quantum gates, analogous to the logic gates of classical computers. For example, an AND gate in a classical setup has output 1 if both its inputs are 0 or 1 – i.e. (0,0) or (1,1). Similarly, a quantum circuit can have qubits and quantum gates wired to combine input values in a certain way.

In such a circuit, a quantum gate could manipulate the qubits to perform specific functions, leading to an output. These outputs can be combined to solve complex mathematical problems.

Classical computers struggle with #P-hard problems – a set of problems that includes estimating the probability that random quantum circuits will yield a certain output.

#P-hard problems are a subset of #P problems, which are all counting problems. To understand what this means, let’s consider another set of problems called NP problems. These are decision-making problems, meaning that the output is always either ‘yes’ or ‘no’.

A famous example of an NP problem is the travelling salesman problem. Given a set of cities, is there a route passing through all of them and returning to the first one, without visiting any city twice, whose total distance is less than a certain value? As the number of cities increases, the problem becomes vastly more difficult to solve.

To turn this NP problem into a #P problem, we must count all the different possible routes that are shorter than the specified limit. #P problems are at least as hard as NP problems because they require not just a ‘yes’ or ‘no’ answer but the number of possible solutions. That is, when the answer is ‘no’, the count will be zero; but when the answer is ‘yes’, the count will have to be computed.

If a problem is #P-hard, then it is so challenging that if you can efficiently solve it, you can also efficiently solve every other problem in the #P class by making certain types of transformations.

Taking the Cayley path

To prove that there is a class of problems that can be solved by quantum computers but not by classical computers, Dr. Movassagh used a mathematical construct called the Cayley path.

The Cayley path is like a bridge that helps the travelling salesman move smoothly between two different situations in the study – like one random route and one significantly complicated route. With quantum computers, one situation would be the worst-case scenario, like imagining the most challenging quantum circuit possible. The other would be the average case, a quantum circuit that has been randomly selected from the set of all possible circuits.

This ‘bridge’ allows us to reframe the most challenging quantum circuit in terms of the average circuit – like seeing how tough it might be to handle the worst traffic jam compared to your regular commute.

Dr. Movassagh showed that estimating the output probability of a random quantum circuit is a #P-hard problem, and has all the characteristics of a problem in this computational complexity class – including overwhelming the ability of a classical computer to solve it.

His paper is also notable because of its error-quantifiable nature. That is, the work dispenses with approximations, and allows independent researchers to explicitly quantify the robustness of his findings.

Quantum complexity theory

As such, Dr. Mossavagh’s paper shows that there exists a problem that presents a computational barrier to classical computers but not to quantum computers (assuming a quantum computer can crack a #P-hard problem).

The establishment of quantum supremacy will have a positive impact on several fields: cryptography is expected to be a particularly famous beneficiary, at least once the requisite advances in hardware and materials science have been achieved.

Dr. Movassagh’s paper is also an advance in quantum complexity theory. The sets NP, #P, #P-hard, etc. were defined keeping the computational abilities of classical computers in mind. Quantum complexity theory is concerned with limits of complexity defined by quantum computers.

The theory also challenges the extended Church-Turing thesis, which is the idea that classical computers can efficiently simulate any physical process. Dr. Movassagh hopes to continue his work to investigate the hardness of additional quantum tasks and someday disprove the thesis.

Tejasri Gururaj is a freelance science writer and journalist.

Quantum computing is becoming more popular – both as a field of study and in the public imagination. The technology promises more speed and more efficient problem-solving abilities, challenging the boundaries set by classical, conventional computing.

The hype has led to inflated expectations. But whether or not it can meet them, the raison d’être of a quantum computer is taken to be synonymous with the ability to solve some problems much faster than a classical computer can. This achievement, called quantum supremacy, will establish quantum computers as superior machines.

Scientists have been exploring both experimental and theoretical ways to prove quantum supremacy.

Ramis Movassagh, a researcher at Google Quantum AI, recently had a study published in the journal Nature Physics. Here, he has reportedly demonstrated in theory that simulating random quantum circuits and determining their output will be extremely difficult for classical computers. In other words, if a quantum computer solves this problem, it can achieve quantum supremacy.

But why do such problems exist?

Facing the quantum challenge

Quantum computers use quantum bits, or qubits, whereas classical computers use binary bits (0 and 1). Qubits are fundamentally different from classical bits as they can have the value 0 or 1, as a classical bit can, or a value that’s a combination of 0 and 1, called a superposition.

Superposition states allow qubits to carry more information. This capacity for parallelism gives quantum computers their archetypal advantage over classical computers, allowing them to perform a disproportionately greater number of operations.

Qubits also exhibit entanglement, meaning that two qubits can be intrinsically linked regardless of their physical separation. This property allows quantum computers to tackle complex problems that may be out of reach of classical devices.

All this said, the real breakthrough in quantum computing is scalability. In classical computers, the processing power grows linearly with the number of bits. Add 50 bits and the processing power will increase by 50 units. So the more operations you want to perform, the more bits you add.

Quantum computers defy this linearity, however. When you add more qubits to a quantum computer, its computational power for certain tasks grows exponentially as 2ⁿ, where n is the number of qubits. For example, whereas a one-qubit quantum computer can perform 2¹ = 2 computations, a two-qubit quantum computer can perform 2² = 4 computations, and so forth.

#P-hard problems

Quantum circuits are at the heart of quantum computing. These circuits consist of qubits and quantum gates, analogous to the logic gates of classical computers. For example, an AND gate in a classical setup has output 1 if both its inputs are 0 or 1 – i.e. (0,0) or (1,1). Similarly, a quantum circuit can have qubits and quantum gates wired to combine input values in a certain way.

In such a circuit, a quantum gate could manipulate the qubits to perform specific functions, leading to an output. These outputs can be combined to solve complex mathematical problems.

Classical computers struggle with #P-hard problems – a set of problems that includes estimating the probability that random quantum circuits will yield a certain output.

#P-hard problems are a subset of #P problems, which are all counting problems. To understand what this means, let’s consider another set of problems called NP problems. These are decision-making problems, meaning that the output is always either ‘yes’ or ‘no’.

A famous example of an NP problem is the travelling salesman problem. Given a set of cities, is there a route passing through all of them and returning to the first one, without visiting any city twice, whose total distance is less than a certain value? As the number of cities increases, the problem becomes vastly more difficult to solve.

To turn this NP problem into a #P problem, we must count all the different possible routes that are shorter than the specified limit. #P problems are at least as hard as NP problems because they require not just a ‘yes’ or ‘no’ answer but the number of possible solutions. That is, when the answer is ‘no’, the count will be zero; but when the answer is ‘yes’, the count will have to be computed.

If a problem is #P-hard, then it is so challenging that if you can efficiently solve it, you can also efficiently solve every other problem in the #P class by making certain types of transformations.

Taking the Cayley path

To prove that there is a class of problems that can be solved by quantum computers but not by classical computers, Dr. Movassagh used a mathematical construct called the Cayley path.

The Cayley path is like a bridge that helps the travelling salesman move smoothly between two different situations in the study – like one random route and one significantly complicated route. With quantum computers, one situation would be the worst-case scenario, like imagining the most challenging quantum circuit possible. The other would be the average case, a quantum circuit that has been randomly selected from the set of all possible circuits.

This ‘bridge’ allows us to reframe the most challenging quantum circuit in terms of the average circuit – like seeing how tough it might be to handle the worst traffic jam compared to your regular commute.

Dr. Movassagh showed that estimating the output probability of a random quantum circuit is a #P-hard problem, and has all the characteristics of a problem in this computational complexity class – including overwhelming the ability of a classical computer to solve it.

His paper is also notable because of its error-quantifiable nature. That is, the work dispenses with approximations, and allows independent researchers to explicitly quantify the robustness of his findings.

Quantum complexity theory

As such, Dr. Mossavagh’s paper shows that there exists a problem that presents a computational barrier to classical computers but not to quantum computers (assuming a quantum computer can crack a #P-hard problem).

The establishment of quantum supremacy will have a positive impact on several fields: cryptography is expected to be a particularly famous beneficiary, at least once the requisite advances in hardware and materials science have been achieved.

Dr. Movassagh’s paper is also an advance in quantum complexity theory. The sets NP, #P, #P-hard, etc. were defined keeping the computational abilities of classical computers in mind. Quantum complexity theory is concerned with limits of complexity defined by quantum computers.

The theory also challenges the extended Church-Turing thesis, which is the idea that classical computers can efficiently simulate any physical process. Dr. Movassagh hopes to continue his work to investigate the hardness of additional quantum tasks and someday disprove the thesis.

Tejasri Gururaj is a freelance science writer and journalist.

Information technology (IT) has become essential to communication, banking, business, health, education, entertainment, and many other walks of our lives. Its prevalence makes us wonder if society can survive without it. IT relies on gadgets that store and process vast amounts of information at humanly impossible speeds.

Gate in computing

A bit is the smallest piece of information storage (it is a portmanteau of binary digit). Often, a large number of bits is required to convey meaningful information. With the advent of modern semiconductor technology, we routinely speak of household computers having a few terabytes (8 trillion bits) of information storage. One terabyte can store 500 hours of high-definition video content.

In a computer, a bit is a physical system with two easily discernible configurations, or states – e.g. high and low voltage. These physical bits are useful to represent and process expressions that involve 0s and 1s: for instance, low voltage can represent 0 and high voltage can represent 1.

A gate is a circuit that changes the states of bits in a predictable way. The speed at which these gates work determines how fast a computer functions.

The quantum gate

Modern computers use semiconductor transistors to build circuits that function as bits. A semiconductor chip hosts more than 100 million transistors on 1 sq. mm. Imagine how small an individual transistor is and how close it is to adjacent transistors. As transistors become smaller, they become more susceptible to quantum effects. This is not desirable as the existing technology will then become unreliable for computational tasks. So there is a limit to how many transistors a computer can have.

Moore’s law, announced in 1965, states that computing power increases tenfold every five years. This law no longer holds as we have already slowed to a two-fold increase every five years. But this doesn’t have to mean we are nearing the end of computing development: the quantum revolution is coming.

The most basic unit of a quantum computer is a quantum bit, or qubit. Like in a conventional computer, it is a physical object that has two states. For example, the spin of a particle can point along two different directions, so the particle can function as a qubit. Or it can be a superconducting circuit that mimics an atom, and its two states can be a ground state, where it has lower energy, and a higher ‘excited’ state.

A quantum gate is a physical process or circuit that changes the state of a qubit or a collection of qubits.

In the quantum-computing context, if particles or superconducting qubits are the physical qubits, the gate is often an electromagnetic pulse.

Interlude: Superposition

A fundamental limitation of conventional computing architecture is that each bit can exist in only one of the two states, 0 or 1. But according to quantum physics, a qubit can also be in a superposition of its two states at the same time.

Imagine you’re walking in the northeast direction. It is equivalent to moving partly along the north and the rest along the east. Your northeast movement is a superposition of walking along the north and along the east. So by combining different distances along the two directions, you can realise some movement in any direction between the two.

The basis states of the qubit are similar to the north and east directions. A qubit in a superposition has some contributions from each basis state. Different superpositions correspond to different amounts of contributions.

If a qubit is in a superposition, then measuring the qubit will cause it to collapse to one of the two states (i.e. either north or east). However, we can only predict the probability that it will collapse to one state. Quantum computers use this to their advantage.

For example, to perform one calculation that requires 16 different inputs, a classical computer requires a total of four bits and sixteen computations. But with four qubits in superposition, a quantum computer could generate answers corresponding to all 16 inputs in a single computation.

Superposition is one of the main factors responsible for speeding up a quantum computer.

But while superposition provides enormous advantages, it is a fragile effect. It deteriorates when qubits interact with their environment. Identifying ways to sidestep or overcome this fragility is an active area of research today.

What gates do

In quantum computers, quantum gates act on qubits to process information. For example, a quantum NOT gate changes the state of a qubit from 0 to 1 and vice versa. The effect of the NOT gate on a superposition is again a superposition, resulting from the action of the NOT gate on each basis state in the initial superposition.

Notably, this feature is common to all quantum gates: the effect of a quantum gate on a superposition is the superposition of the effects of the quantum gate on the basis states contributing to the initial superposition.

So as the quantum NOT gate inter-converts the states 0 and 1, its action is to swap the contributions of the basis states in the superposition.

The Hadamard gate is a type of gate that acts on a single qubit: it generates a superposition of the basis states.

The controlled-NOT, or CNOT, gate acts on two qubits: a control qubit and a target qubit. The control qubit is unaffected by the CNOT gate. The target qubit flips from 0 to 1 or 1 to 0 if the control qubit state is 1.

CNOT plus a few other gates (that act on single qubits) can perform all possible logical operations on binary information encoded on qubits. That is, they can be combined to form quantum circuits capable of processing information.

Research on reliable quantum computers and suitable quantum algorithms is happening in many institutes, universities, and research labs worldwide. Large-scale, reliable quantum computers will benefit industries ranging from drug design to safe communications.

S. Srinivasan is a professor of physics at Krea University.