The Oracle Separation of BQP and PH: A Recent Advancement in Quantum Complexity Theory A CS254 STUDENT Classical computing enables us to encode computation as transitions between of Turing Machines [7], we will not discuss them here. Instead, we concrete memory states according to a set of rules. Quantum computing will present some fundamental quantum gates, which can be com- harvests the power of quantum mechanics to allow us to compute with posed together to describe the full space of quantum programs. In superpositions (i.e. a linear combination) of memory states. We gain a de- the following sections, I will describe how qubits are represented as gree of parallelism that we can exploit for problems exhibiting a certain vectors in C2, how computational states are represented as tensor structure e.g. factoring numbers. This paper presents a brief overview of the products of qubits (and more compactly, in Dirac bra-ket notation), theoretical underpinnings of quantum computing and quantum complexity theory without assuming any background in quantum physics or quantum and how gates are represented as unitary matrices. Before discussing computing. We will then focus on a recent breakthrough in the field: the ora- the main result of the Raz Tal paper, I will also provide an example of cle separation of the complexity classes BQP (bounded quantum polynomial two problems that can be solved exponentially faster on a quantum time) and PH (polynomial hierarchy). Understanding this result will give us computer as compared to a classical computer: the Deutsch Oracle a deeper appreciation for the new problems we are able to solve efficiently Problem and Simon’s problem (which was a precursor to Shor’s on quantum computers but not on their classical counterparts. famous algorithm for factoring). Note that the following is not a Additional Key Words and Phrases: Quantum Computing, BQP, PH, Com- comprehensive review of quantum computing, although hopefully plexity Theory, Oracles it will provide the reader with guidance in being able to look up further resources as necessary. ACM Reference Format: A CS254 Student. 2019. The Oracle Separation of BQP and PH: A Recent Advancement in Quantum Complexity Theory. 1, 1 (March 2019), 5 pages. 2 QUBITS https://doi.org/10.1145/nnnnnnn.nnnnnnn As column vectors, the qubits for 0 and 1 are represented as follows: 1 INTRODUCTION 1 0 j0i = ; j1i = Most modern computers harvest the power of transistors as a cheap 0 1 and scalable way to represent memory and operations on memory. The j·i is called ket notation. We can describe states of multiple Logical gates (e.g. AND, OR, NOT, NAND, NOR, XOR) enable us qubits using ket notation. The computational state of multiple qubits to perform operations on these memory states. Turing machines is their tensor product: and Boolean circuits capture this kind of classical computation well since we deal only with concrete memory states. 203 203 Memory states in a quantum computer take on a more probabilis- 6 7 6 7 617 607 tic nature. Instead of a bit being 0 or 1, they can be somewhere in j01i = j0i ⊗ j1i = 6 7 ; j3i = j11i = j1i ⊗ j1i = 6 7 607 607 between – collapsing to one of these states with a certain probability 6 7 6 7 607 617 dependent on the physical system that represents the bit. Think of 4 5 4 5 the classic Schrodinger’s Cat thought experiment – until you open As mentioned before, the components of these vectors can take on the box, the cat is both dead and alive (each with a probability of complex values. In this representation, the sums of the squares of 50%). Similarly, we can place quantum bits, known as qubits in a the components of a vector describing a qubit evaluates to 1. We can state of superposition where the qubit can collapse to a 0 or a 1, write a qubit in the form a0 j0i +a1 j1i, we have that the probability 2 each with a controllable probability, once measured by detectors. the qubit collapses to j0i when measured is ja0j . Similarly, the 2 These qubits can be represented by various physical systems and probability the qubit collapses to j1i when measured is ja1j . phenomena e.g. (polarization?) photos, electron spins, etc. Often, two kets may be written together to implicitly denote their Since our fundamental unit of representation, the qubit, has this tensor product i.e. j0i j1i = j0i ⊗ j1i = j01i. strange probabilistic caveat, we must seek a different kind of theo- retical representation. Although there do exist quantum analogues 3 QUANTUM GATES Author’s address: A CS254 Student. We often represent Quantum gates as matrices since they can be viewed as linear transformations acting upon qubit states, which Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed can be viewed as vectors. On a more technical note, a matrix U that for profit or commercial advantage and that copies bear this notice and the full citation represents a quantum gate must be unitary and have the property on the first page. Copyrights for components of this work owned by others than ACM y y must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, U U = I, where U denotes the complex conjugate transpose of to post on servers or to redistribute to lists, requires prior specific permission and/or a U . This detail has to do with preserving the nice mathematical fee. Request permissions from [email protected]. properties of qubits we discussed above as well as the physical © 2019 Association for Computing Machinery. XXXX-XXXX/2019/3-ART $15.00 nature of qubits. In the following sections, we will discuss several https://doi.org/10.1145/nnnnnnn.nnnnnnn common quantum gates. , Vol. 1, No. 1, Article . Publication date: March 2019. 2 • A CS254 Student 3.1 NOT We can extend the Hadmard operator to n qubits as follows: the 1 Í x ·y Suppose we wish to define a linear operator that maps j0i to j1i and Hadamard of jx1i jx2i ::: jxni is p (−1º jyi. Where j1i to j0i i.e. performs negations on qubits. We could express this 2n y 2f0;1gn transformation as the following matrix: ¹·º denotes bitwise dot-product mod 2. This formula follows from 1 x 0 1 the expansion of the tensor product p ¹j0i + (−1º 1 j1iº¹j0i + X = 2n 1 0 (−1ºx2 j1iº · · · ¹j0i + (−1ºxn j1iº, which follows from the fact that 1 We denote this matrix as X because it is also known as the Pauli- we can consider a Hadamard applied to a single qubit x as p ¹j0i + X gate. Using our definitions above, one can verify that indeed 2 1 X j0i = j1i and X j1i = j0i. If we wish to perform a negation (−1ºx j1iº. Note that we must multiply by the p factor for normal- over two qubits, we can apply the tensor product of the operators: 2n ¹X ⊗ Xº j00i = j11i. For an operator Q, we denote Q ⊗ Q ⊗ · · · ⊗ Q ization purposes (recall that the squares of components correspond (n times) as Q ⊗n. to probabilities and the probabilities of all possible states must add Thus far, we have demonstrated operators on concrete qubits. to 1). The real power from this abstraction comes from the fact that we can apply any of these quantum operators over superpositions of 3.5 Oracle Queries states i.e. qubits that can probabilistically be j0i or j1i depending Often times, when discussing complexity results in quantum com- on their underlying wave functions. For instance, we can imagine puting, we consider machines that have access to an oracle that 1 1 can compute some function f : f0; 1gn ! f0; 1g. There are two negating a qubit in the state p ¹j0i − j1iº to obtain p ¹j1i − j0iº, 2 2 common methods of querying this oracle f in a quantum circuit. by linearity. The CNOT-query maps qubits as follows: jx;wi jx;w ⊕ f ¹xºi 3.2 CNOT . The phase-query flips around the input vector if f ¹xº = 1: Building off the idea of a NOT gate, presented in the previous section, E we can obtain a controlled-not (CNOT) gate. The CNOT gate takes ! (− ºf ¹xº jxi 1 x in two qubits as input and negates the second qubit if the first qubit is set to j1i. It has the following matrix representation: 4 BQP 21 0 0 03 Quantum gates give us a foundation with which we can build quan- 6 7 60 1 0 07 tum circuits. A language is in BQP if it can be computed by a poly- 6 7 60 0 0 17 nomial size quantum circuit that errs with a probability of at most 6 7 1 60 0 1 07 . This is analogous to the definition of BPP. Indeed, we can show 4 5 3 3.3 Toffoli that BPP ⊆ BQP [7] since quantum algorithms can use superpo- sition states as coin-flips. BQP essentially captures the problems Further extending the idea of a CNOT gate, we can obtain the Tof- we view as feasible for quantum computers. It is natural to wonder foli gate, which acts on three qubits as follows: whether quantum computers can efficiently solve certain problems jc1i jc2i jti ! jt ⊕ ¹c1 · c2ºi that are infeasible for classical computers. The next two sections That is, the third qubit is flipped if both the first and second qubits describe quantum algorithms that exponentially outperform their are set to j1i.