The motivation for quantum computing is usually argued by its ability to outperform classical computers on specific tasks, giving rise to "quantum supremacy". While speed is undoubtedly a significant advantage of quantum computing, quantum hardware also shows promise in terms of energy efficiency. Because quantum computers can solve some problems with exponential enhancement in speed, it is natural to expect this to translate to reduced energy consumption. Existing classical supercomputers consume significant amounts of power, comparable to the energy consumption of a small town. If these resources are directed at algorithms where quantum advantage exists, one might expect that the quantum advantage lies not only in execution time but also in energy consumption. In this blog post, we discuss the fundamentals of quantum computing from the perspective of *quantum energy advantage* rather than speed advantage.

**What is Quantum Computing, and why is it hard?**

Classical computers employ bits to encode digital information as zeros and ones. Quantum computers, on the other hand, utilize quantum bits or *qubits*. Qubits also encode information into zeros and ones but without the constraint of being strictly one or the other. Instead, a qubit may be an arbitrary *superposition *of the two, where the two values have different weightings, known as *quantum amplitudes*, given by complex numbers. However, these weightings are *not* probabilities but simultaneous. *Entangled* *states* are multi-qubit states whose amplitudes are defined jointly across qubits. The 0 and 1 amplitudes of qubits cannot be described independently but only collectively. Quantum computers achieve advantage by exploiting these new degrees of freedom, which do not exist in classical computers.

Designing and engineering quantum computers is challenging because quantum amplitudes are very fragile. When qubits are measured or interact with their external environment, the quantum amplitudes describing them are irreversibly changed (so-called *wavefunction* *collapse*). As quantum amplitudes encode the information within quantum computers, future large-scale quantum computers will require elaborate error-correction techniques to remain robust against errors, a significant challenge for scaling up existing quantum devices to solve larger-scale problems.

**Quantum Energy Advantage**

Given the diversity of possible physical implementations, estimating the energy consumption of future large-scale quantum computers is difficult. However, qualitative arguments can be made based on *scaling* characteristics, which generally hold without specific implementation assumptions.

Consider a simple classical algorithm, the addition of two *n*-bit integers. This is algorithmically implemented by performing bitwise addition-and-carry operations across all corresponding bits in the two numbers. The number of elementary operations scales linearly with *n*, often written as *O(n)*. Conversely, multiplication requires performing elementary bitwise operations between every pair of bits between the two numbers. So, the complexity of multiplication scales quadratically with *n*, denoted *O(n ^{2}).* The problems of addition and multiplication are very efficient on classical computers, and no quantum advantage can be found.

Some algorithms are much more gruelling than this and exhibit exponential complexity, *O(2 ^{n})*. Under exponential scaling,

*n*does not need to be very large for the problem to become intractable, presenting a fundamental obstacle to executing large instances of such algorithms.

Remarkably, some (but not all) algorithms exhibit exponential complexity scaling when executed on classical computers but more efficient sub-exponential scaling on quantum computers. The goal of quantum algorithm designers is to identify problems at this intersection of classically inefficient yet quantum-efficient. Quantum advantage is found at this intersection.

Now consider a quantum computer efficiently solving a problem that would exhibit inefficient exponential complexity scaling if executed on a classical device. One way to think about this is that the quantum computer is exponentially reducing execution time by comparison –* quantum speed advantage*. Multiplying execution times by the respective power consumption of the quantum and classical devices yields the net quantum and classical energy consumption.

Although we can reasonably expect quantum computers to consume more power than classical computers, given the challenging engineering involved, these scaling relationships dominate the net energy comparison rather than the relative quantum vs classical power consumption. As a simple toy model, consider a scenario where the power consumption of computers is proportional to their size with device-dependent constant factors. Even if these constant factors differ by many orders of magnitude, the exponential complexity scaling difference against *n* dominates the calculation. In simple words, no matter how much more energy it takes to operate a qubit than a bit, it will always be more energy-efficient as long as it solves an exponentially scaling problem.

In a PRX Quantum paper published in 2023, a detailed study of energy consumption for full-stack superconducting quantum computers was conducted (__here__). The quantum energy advantage of cracking *n*-bit RSA keys was estimated. Moreover, in certain parameter regimes, it was shown that quantum computers could achieve energy advantage before demonstrating speed advantage.

Today's quantum computers are small-scale devices (known as NISQ – *Noisy Intermediate Scale Quantum* – devices), lacking the error-correction mechanisms necessary to enable scalability. While today's NISQ devices have claimed to achieve* quantum supremacy*, they are not *universal* (i.e. arbitrarily programmable) quantum computers, rather solving highly specific proof-of-principle problems, which unfortunately have found little real-world utility.

**BTQ's Contribution to Quantum Energy Advantage**

One example of a NISQ-era architecture is *boson-sampling *(__here__), a simple but highly restricted model for optical quantum computing with comparatively simple engineering requirements – an optical interferometer with photon sources and photo-detectors. Boson-samplers have now been demonstrated which convincingly outperform an equivalent classical simulation by many orders of magnitude.

While boson-samplers vastly outperform classical simulators, the problem they solve is a highly restricted one that has found little pragmatic utility.

BTQ researchers recently demonstrated the applicability of boson-sampling to the problem of *proof-of-work *(PoW), a distributed algorithm used in blockchain protocols such as Bitcoin. Proof-of-work is a *consensus algorithm* that aims to randomly choose a small subset of competing nodes in a network to authorize blockchain transactions. The conventional approach to proof-of-work is for nodes to solve computationally intensive inverse-hashing problems, which consume vast amounts of power, a major criticism of current blockchain implementations.

Substituting energy-intensive inverse-hashing with a similarly defined quantum problem based on boson sampling, BTQ researchers demonstrated that a blockchain network comprising quantum nodes performing quantum proof-of-work (QPoW) can achieve the same outcome (distributed consensus) using orders of magnitude less energy than a corresponding network of classical simulators solving the same problem. This advancement demonstrates what might be the first practical use case for boson-sampling and that it affords enormous energy savings for a crucial real-world problem where energy consumption is a significant obstacle.

While the net energy consumption of blockchain networks has numerous contributing factors, many of which are hard to predict or characterize, this research demonstrates the relevance of the quantum/classical divide on some of these considerations and the potential that quantum technology might have in this rapidly advancing sector of the economy. Currently, the Bitcoin network collectively consumes the energy of a medium-sized country (__here__). If quantum energy advantage can be exploited, quantum-enabled blockchains might provide the solution.

See BTQ's recent blog post to learn more about BTQ's quantum solution __here__.