Speculative Technologies

In classical computing, bits encode information as 0s and 1s. In quantum computing, qubits store quantum states, defined as \(2\)-dimensional unit vectors living in complex space. The vector is composed from a linear combination of elementary states, the zero ket (\(\vert{}0\rangle\)) and the one ket (\(\vert{}1\rangle\)). Quantum systems are composed of qubits, wires, and quantum gates. Just as a classical computer can be configurable by different physical systems, quantum systems operate and behave differently using different components, but the underlying principles are the same. A transition through a quantum wire represents a passage in time. Qubits can be represented by any particle, such as a proton, photon, or neutrino. However, some particles are more suited to be qubits than wires and vice versa. Neutrinos have weak interactions with other matter, making it hard for them to manipulate stable systems. Thus, neutrinos are suitable to be used as quantum wires but are poor candidates to be used as qubits.

Max Planck introduced energy quantization in 1900. Einstein expanded the quantum idea to light, explaining the photoelectric effect in 1905. Niels Bohr modeled the atomic structure, where electrons occupied fixed orbits in 1913. Then in 1925, Werner Heisenberg formalized the field with matrix mechanics. Decades later, Alan Turing was trying to prove the Entscheidungsproblem, or the decision problem, proposed by David Hilbert. Building on computing theory, David Deutsch predicted that classical computers cannot simulate quantum processes. He devised the first quantum algorithm that is exponentially faster than any classical algorithm, using a single query instead of two queries using classical means.

When calculating these quantum processes, unitary matrices are essential because they are the only matrices that preserve the length of a vector. Taking the complex transpose of an arbitrary quantum state ket produces a bra, which we denote as \(\langle\psi\vert{}\). The matrix product of a unitary matrix and its Hermitian conjugate, given by the dagger notation, rotates the vector in space but preserves the length of the vector. This is known as Dirac notation, or bra-ket notation. To visualize this concretely, think of how a matrix transforms a vector in space: the operation \(M\vert{}e_k\rangle\) physically maps to the \(k\)-th column of matrix \(M\). Extending this visual logic to pinpointing exact matrix entries, \(M_{jk} = \langle e_j\vert{} M \vert{}e_k\rangle\). In other words, finding the entry of a matrix is the exact equivalent of applying the elementary row vector \(\langle e_j\vert{}\) to the matrix \(M\), followed by applying the column vector \(\vert{}e_k\rangle\). The norm of the ket is the square root of the product of the bra-ket.

Using this mathematical definition, we can venture into quantum gates involving two qubits. The geometric intuition for applying quantum gates on qubits is the rotation of the state vector on the 3D surface of a sphere called the Bloch Sphere. The computational basis states for a 2-qubit system are \(\vert{}00\rangle\), \(\vert{}01\rangle\), \(\vert{}10\rangle\), and \(\vert{}11\rangle\). The CNOT gate is the quantum equivalent of the NOT gate for two qubits. This flipping mechanism is used as follows: define the control qubit and target qubit. Suppose \(\vert{}x\rangle\) is the control and \(\vert{}y\rangle\) is the target. The gate transforms the state such that \(\vert{}xy\rangle \rightarrow \vert{}x, y \oplus x\rangle\). This means we are modulo the addition of \(x\) and \(y\). If \(x\) is 0, the switch is off, thus the state of \(y\) doesn’t change. If \(x\) is 1, then \(y\) flips. Next is the Toffoli gate, the equivalent of the AND gate in classical computing. Because qubits have to be unitary and reversible, \((x \land y)\) is very difficult to implement in quantum computing. The NAND gate can be constructed by taking apart the \(z\) component of the Toffoli gate and replacing \(z\) with 1. Ultimately, quantum computing consists of three parts: the input being the initial computational basis state, passing through a quantum circuit composed of CNOT gates and other quantum gates, whose output is measured by a classical bit.

The holy grail of quantum computing is to simulate any physical system, though simulating the universe remains incredibly hard because it is complex, messy, and nondeterministic. So far, physicists are unable to bridge the gap between the standard model and Einstein’s general relativity, though John Preskill and his collaborators have made progress using quantum computing to simulate quantum field theory. In this realm, quantum systems could potentially simulate subatomic particle interactions and collisions, guide smart materials to change form on command, or coordinate tiny systems to build or fix objects.

Furthermore, atom behavior at the quantum level is very poorly simulated by chemists, making drug discovery slow. In drug discovery, quantum algorithms could model chemical bonds and atoms, speed up tests for drug compounds, and cut down trial and error costs in labs.

On the geopolitical front, intelligence agencies have been interested in the development of quantum computing to crack encryptions since the mid-1990s in hopes of gathering intelligence. An proven result of quantum computing is Shor’s quantum factorization algorithm devised by American theoretical computer scientist Peter Shor. Shor's algorithm can factorize any positive real number, in the hundreds of digits, to its prime factors in polynomial time versus the best classical algorithms that can only do so in sub-exponential time. Theoreticially it can unlock widely used security keys and force global organizations to adopt quantum-safe encryption, yet to be practically useful, breaking real-world secure encryption keys, such as 2048-bit RSA key, requires millions of qubits. Quantum decoherence and high error rates have contained progress on the number of qubits from 2 in 1997 (IBM, Oxford, UC Berkeley, Stanford, MIT) to 120 (IBM Nighthawk 2025) over the past 20 years. While quantum physicists can agree on the underlying math, physical interpretations vastly differ, and the societal implications are much more uncertain. Even though large-scale fault-tolerant quantum computers remain decades away, our security risks and economic inequalities remain heavily dependent on the proactive migration to post-quantum cryptography today.

• What are suitable objects for qubits?
• Who developed quantum theory in the 1900s to 1925?
• How did each contribute to our understanding of quantum mechanics?
• What problem was Alan Turing trying to solve from the mathematician David Hilbert?
• What problem did David Deutsch predict, and how did he formalize a universal computing algorithm?
• What can quantum physicists agree on? Where do interpretations vastly differ?
• What quantum algorithms could potentially be useful that classical computers fail at?
• What can quantum computers do to advance drug discovery, discover new chemical elements, break encryption for world’s intelligence agencies, simulate quantum field theories, and programmable matter?
• What are the tradeoffs of working for the government, in the intelligence agency, complaining about contributing to the advancement of surveillance states and the invasion of privacy, while on the other hand, taking the money with some self-serving justification that they’re not really helping? (Ronald de Wolf - The Potential Impact of Quantum Computers on Society)
• Why do quantum physicists care about unitary matrices when it comes to quantum computing?
• How does the CNOT gate actually work? And why is the Toffoli gate (AND) so hard to implement?
• What is entanglement in precise mathematical terms?
• What is the geometric intuition for thinking about applying quantum gates on qubits?
• Why are unitary matrices the only matrices which preserve length?
• Why is it so hard to simulate the universe?